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University  of  California. 

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LELAND  STANFORD  JUNIOR  UNIVERSITY 

STANFORD  UNIVERSITY,  P.   O.,   CAL. 


MECHANICAL   ENGINEERING 


-'II        /\  ^  • 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementarymachinOOsmitrich 


ELEMENTARY 


MACHINE     DESIGN 


ALBERT   W.  SMITH 


Profeaaor  of  Mechanical  Engineering 
Leland  Stanford  Jr.  Utiiversity 


CALIFORNIA 

STANFORD   UNIVERSITY   PRESS 

1895 


Copyrighted,  1894,  by  A.  W.  Smith, 


ERRATA  FOR  P:LEMENTARY  MACHINE  DESIGN. 


Page  18,  tifth  line,  for  "  r  "  read  "  (/." 

"     19,  first  line,  before  "The  centro ''  insert  "Fig.  19." 

"     22,  twelfth  line  from  bottom,  after  "diagram,"  insert  "transfer 

to  a  'time  base'  as  on  page  38." 

u     o<    +        ^      ■   4.U  ^'       ■         ^  i.1.  •       ,, max.  Fi  of  slider  „ 

24,  twentv-sixth  line,  invert  the  expression  " " 

[  I  of  ab 

Fig.  28,  reverse  arrow  that  indicates  the  direction  of  rotation. 

"  24,  indicate  the  intersection  of  a  and  e  by  "  T." 

Page  27,  first  line,  for  "  cd  "  read  "  af.  " 

"  80,  fifteenth  line,  for  "j"  read  "2." 

"  82,  eighteenth  line,  for  "revolutions"  read  "strokes." 

"  40,  sixth  line  from  bottom,  for  "38"  read  "87." 

Fig.  40,  indicate  arc  of  small  circle  between  he  and  M  by  "/)." 
Page  44,  twelfth  line,  for  "  C/;"  read  "  TD." 

"  47,  first  line,  for  "45"  read  "44." 

"  68,  first  part  of  seventh  line  from  bottom,  for  "  Po  "  read  "8." 

"  78,  fifteenth  line,  for  "  C"  read  "  0." 

"  84,  third  line,  for  "  Sin  "  read  "  Tan." 

"  86,  fifth  line,  for  "sin"  read  "tan." 

"  90,  ninth  line,  for  "  7\  +  T,"  read  "  T,  —  T,.'' 

"  91,  eighth  line,  after  n  insert  "  Fig.  96." 

"     98,  twentieth  line,  for  "— (  )"  read  "— ^(  )." 

'•     98,  in  equations  "  lf^= "insert   "2"   as  a   factor    in    the 

numerator  of  the  right-hand  member,  and  make  corre- 
sponding change  in  result. 

"     158,  eighth  line,  for  "Z>"  read  "  0." 

[USIVBRSIIT] 


Professor  John  E.  Sweet 


THIS    book    is    dedicated 


AS    AN     EXPRESSION    OF    AFFECTION,     AND    IN 
ACKNOWLEDGMENT    OF    YEARS    OF    HELPFULNESS. 


'UHIVEKSITT] 

PREFACE 


One  can  never  become  a  machine  designer  by  studying  a  book. 
The  true  designer  is  one  whose  judgment  is  ripened  by  experience 
in  constructing  and  operating  machines.  Mr.  William  B.  Bement, 
a  designer  in  the  true  sense,  once  said  that  he  thought  it  useful  to 
figure  the  strength  of  machine  parts,  because  the  results  were  sug- 
gestive to  the  designer.  One  may  know  thoroughly  the  laws  which 
govern  the  transmission  of  energy;  may  understand  much  concern- 
ing the  nature  of  constructive  materials  ;  may  know  how  to  obtain 
results  by  mathematical  processes  ;  and  yet  be  unable  to  design  a 
good  machine.  One  needs  also  to  know  the  thousand  and  one  things 
connected  with  practice,  which  constantly  modify  design,  so  that  one 
can  take  the  results  of  computation  and  accept,  reject,  and  modify, 
until  the  machine  will,  when  constructed,  do  its  required  work 
satisfactorily  and  enduringly.  The  writer  once  heard  Professor 
John  E.  Sweet  say  :  "  It  is  comparatively  easy  to  design  a  good 
new  machine,  but  it  is  very  hard  to  design  a  machine  which  will  be 
good  when  it  is  old."  This  quality  of  foresight  only  comes  with 
long  experience. 

There  is,  however,  a  certain  part  of  the  designer's  mental  equip- 
ment which  may  be  furnished  in  the  class-room,  or  by  books. 
This  is  the  writer's  excuse  for  the  following  pages. 

Even  Elementary  Machine  Design  cannot  be  treated  exhaust- 
ively. The  kinds  of  machines  are  too  numerous,  and  their  differ- 
ences are  too  great.  An  effort  is  made  here  to  suggest  methods  of 
reasoning,  rather  than  to  give  rules.  A  knowledge  of  the  usual 
university  course  in  pure  and  applied  mathematics  is  presupposed. 


VI  PREFACE. 

The  part  upon  "  Motion  in  Machines "  could  not  have  been 
written  without  the  use  of  the  excellent  book  "The  Mechanics  of 
Machinery,"  by  Prof.  A.  B.  W.  Kennedy.  To  him  and  to  Prof.  L. 
M.  Hoskins,  to  whom  the  writer  has  so  often  gone  for  the  help  which 
never  failed,  grateful  acknowledgment  is  here  made.  In  several 
places  acknowledgment  is  made  to  others  ;  yet  the  writer  feels  that 
he  has  failed,  though  unintentionally,  to  give  credit  for  much  of 
the  best  he  has  received. 

A.  \y.  S. 

Stanford  University,  Califorxia,  January,  1895. 


[USIVBRSITT] 

CONTENTS. 

PAGE 

PltEFACE  ...........         V 

INTRODUCTION .  ix 

CHAl^ER  I. 

PRELIMINARY  ..........  1 

CHAPTER    II. 
MOTION    IN    MECHANIBMt^     .  .  .  .  .  .  .  .  11 

CHAPTER   in. 
ENERGY    IN    MACHINES 28 

CHAPTER   IV. 

PARALLEL    OR   STRAIGHT    LINE    MOTIONS      .....  B7 

CHAPTER    V. 
TOOTHED    WHEELS,    OK    GEARS  .  .  .  .  .  .  .39 

CHAl^'ER  VI. 
CAMS 72 

CHAPTER   VII. 

BELTS 75 

CHAPTER   VIII. 

DESIGN    OF    FLY-WHEELS 93 

CHAPTER  IX. 

RIVETED    JOINTS 100 


Vlll  CONTENTS. 

PAGE 

CHAPTER  X. 
DESIGN    OF   JOURNALS  . 110 

CHAPTER  XI. 
SLIDING    SURFACES  . 126 

CHAPTER  XII. 

BOLTS   AND   SCREWS    AS   MACHINE   FASTENINGS     ....  130 

CHAPTER  XIII. 

MEANS  FOR  PREVENTING  RELATIVE  ROTATION    .     .     .     .137 

CHAPTER  XIV. 

FORM    OF   PARTS   AS    DICTATED   BY    STRESS  .  .  .  .140 

CHAPTER  XV. 
MACHINE   SUPPORTS .  ,  .146 

CHAPTER  XVI. 
MACHINE   FRAMES       .  .  ., 150 

INDEX 161 


^i?-=^^^^ 


[TJFIVBRSITT 

INTRODUCTION 


In  general  there  are  four  considerations  of  prime  importance  in 
designing  machines :  I.  Adaptation,  II.  Strength  and  Stiffness, 
III.  Economy,  IV.  Appearance. 

^  I.  This  requires  all  complexity  to  be  reduced  to  its  lowest  terms 
in  order  that  the  machine  shall  accomplish  the  desired  result  in  the 
most  direct  wa}^  possible,  and  with  greatest  convenience  to  the  op- 
erator. 

II.  This  requires  the  machine  parts  subjected  to  the  action  of 
forces  to  sustain  these  forces,  not  only  without  rupture,  but  also  with- 
out such  yielding  as  would  interfere  with  the  accurate  action  of  the 
machine.  In  many  cases  the  forces  to  be  resisted  may  be  calculated, 
and  the  laws  of  Mechanics,  and  the  known  qualities  of  constructive 
materials  become  factors  in  determining  proportions.  In  other 
cases  the  force,  by  the  use  of  a  "  breaking  piece,"  may  be  limited  to 
a  maximum  value,  which  therefore  dictates  the  design.  But  in 
many  other  cases  the  forces  acting  are  necessarily  unknown  ;  and 
appeal  must  be  made  to  the  precedent  of  successful  practice,  or  to 
the  judgment  of  some  experienced  man,  until  one's  own  judgment 
becomes  trustworthy  by  experience. 

In  proportioning  machine  parts,  the  designer  must  always  be 
sure  that  the  stress  which  is  the  basis  of  the  calculation  or  the  esti- 
mate, is  the  maximum  possible  stress.  Otherwise  the  part  will  be 
incorrectly  proportioned.  For  instance,  if  the  arms  of  a  pulley 
were  to  be  designed  solely  on  the  assumption  that  they  endure  only 
the  transverse  stress  due  to  the  belt  tension,  they  would  be  found 
to  be  absurdly  small,  because  the  stresses  resulting  from  the  shrink- 


X  INTRODUCTION. 

age  of  the  casting  in  cooling,  are  often  far  greater  than  those  due 
to  the  helt  pull. 

The  design  of  many  machines  is  a  result  of  what  may  he  called 
"machine  evolution."  The  first  machine  was  built  according  to 
the  best  judgment  of  its  designer  ;  but  that  judgment  was  fallible, 
and  some  part  yielded  under  the  stresses  sustained  ;  it  was  replaced 
by  a  new  part  made  stronger  ;  it  yielded  again,  and  again  was  en- 
larged, or  perhaps  made  of  some  more  suitable  material  ;  it  then 
sustained  the  applied  stresses  satisfactorily.  Some  other  part 
yielded  too  much  under  stress,  although  it  was  entirely  safe  from 
actual  rupture  ;  this  part  was  then  stiffened,  and  the  process  con- 
tinued, till  the  whole  machine  became  properly  proportioned  for 
the  resisting  of  stress.  Manj^  valuable  lessons  have  been  learned 
from  this  process  ;  many  excellent  machines  have  resulted  from  it. 
There  are,  however,  two  objections  to  it :  it  is  slow  and  very  expen- 
sive, and  if  any  part  had  originally  an  excess  of  material,  it  is  not 
changed  ;  only  the  parts  that  yield  are  perfected. 

III.  The  attainment  of  economy  does  not  necessarily  mean  the 
saving  of  metal  or  labor,  although  it  may  mean  that.  To  illustrate: 
Suppose  that  it  is  required  to  design  an  engine  lathe  for  the  market. 
The  competition  is  sharp ;  the  profits  are  small.  How  shall  the 
designer  change  the  design  of  the  lathes  on  the  market  to  increase 
profits  ?  (a)  He  may,  if  possible,  reduce  the  weight  of  metal  used, 
maintaining  strength  and  stiffness  by  better  distribution.  But  this 
must  not  increase  labor  in  the  foundry  or  machine  shop,  nor  reduce 
weight  which  prevents  undue  vibrations.  (b)  He  may  design 
special  tools  to  reduce  labor  without  reduction  of  the  standard  of 
workmanship.  The  interest  on  the  first  cost  of  these  special  tools, 
however,  must  not  exceed  the  possible  gain  from  increased  profits, 
(c)  He  may  make  the  lathe  more  convenient  for  the  workmen. 
True  economy  permits  some  increase  in  cost  to  gain  this  end. 
It  is  not  meant  that  elaborate  and  expensive  devices  are  to  be  used, 
such  as  often  come  from  men  of  more  inventiveness  than  judgment, 
and  which  usually  find  their  level  in  the  scrap  heap  ;  but  that  if  the 
parts  can  be  rearranged,  or  in  any  way  changed  so  that  the  lathes- 


INTRODUCTION.  XI 

man  shall  select  this  lathe  to  use  because  it  is  handier,  when  other 
lathes  are  available,  then  economy  has  been  served,  even  though 
the  cost  has  been  somewhat  increased  ;  because  the  favorable  opin- 
ion of  intelligent  workmen  means  increased  sales. 

In  (a)  economy  is  served  by  a  reduction  of  metal  ;  in  (b)  by  a 
reduction  of  labor  ;  in  (c)  it  may  be  served  by  an  increase  of  both 
labor  and  material. 

The  addition  of  material  largely  in  excess  of  that  necessary  for 
strength  and  rigidity,  to  reduce  vibrations,  may  also  be  in  the 
interest  of  economy,  because  it  may  increase  the  durability  of  the 
machine  and  its  foundation  ;  may  reduce  the  expense  incident  upon 
repairs  and  delays,  thereby  bettering  the  reputation  of  the  machine, 
and  increasing  sales. 

Suppose,  to  illustrate  further,  that  a  machine  part  is  to  be 
designed,  and  either  of  two  forms,  A  or  B,  will  serve  equally  well. 
The  part  is  to  be  of  cast  iron.  The  pattern  for  A  will  cost  twice  as 
much  as  for  B.  In  the  foundry  and  machine  shop,  however,  A 
can  be  produced  a  very  little  cheaper  than  B.  Clearly  then,  if  but 
one  machine  is  to  be  built,  B  should  be  decided  on  ;  whereas,  if  the 
machine  is  to  be  manufactured  in  large  numbers,  A  is  preferable. 
Expense  for  patterns  is  a  Urst  cost.  Expen^se  for  work  in  the  foun- 
dry and  machine  shop  is  repeated  with  each  machine. 

Economy  of  operation  also  needs  attention.  This  depends  upon 
the  efficiency  of  the  machine ;  i.  e.,  upon  the  proportion  of  the 
energy  supplied  to  the  machine  which  really  does  useful  work.  This 
efficiency  is  increased  by  the  reduction  of  useless  friction  a  1  resist- 
ances, by  careful  attention  to  the  design  and  means  of  lubrication, 
of  rubbing  surfaces. 

In  order  that  economy  may  be  best  attained,  the  machine 
designer  needs  to  be  familiar  with  all  the  processes  used  in  the 
construction  of  machines  —  pattern  making,  foundry  work,  forging, 
and  the  processes  of  tlie  machine  shop  —  and  must  have  them  con- 
stantly in  mind,  so  that  while  each  part  designed  is  made  strong 
enough  and  stiff  enough,  and   properly  and  conveniently  arranged, 


Xll  .  INTRODUCTION. 

and  of  such  form  ae  to  be  satisfactory  in  appearance,  it  also  is 
80  designed  that  the  cost  of  construction  is  a  minimum, 

ly.  The  fourth  important  consideration  is  Appearance.  There 
is  a  beauty  possible  of  attainment  in  the  design  of  machines  which  ie 
always  the  outgrowth  of  a  purpose.  Otherwise  expressed  ;  A  ma- 
chine to  be  beautiful  must  he  purposeful.  Ornament  for  ornament's 
sake  is  seldom  admissible  in  machine  design.  And  yet  the  striving 
for  a  pleasing  effect  is  as  much  a  part  of  the  duty  of  the  rriachine 
designer  as  it  is  a  part  of  the  duty  of  an  architect. 


ELEHENTARY    HACHINE    DESIGN. 


CHAPTER   L 


PRELIMINARY. 


1.  The  solution  of  problems  in  machine  design  involves  the 
consideration  of  force,  motion,  work,  and  energy.  It  is  assumed 
that  the  student  understands  clearly  what  is  meant  by  these  terms. 

A  complete  cycle  of  action  of  a  machine  is  such  an  interval 
that  all  conditions  in  the  machine  are  the  same  at  its  beginning 
and  end. 

The  law  of  Conservation  of  Energy  underlies  every  machine  prob- 
lem. This  law  may  be  expressed  as  follows  :  The  sum  of  energy 
in  the  universe  is  constant.  Energy  may  be  transferred  in  space; 
it  may  be  changed  from  one  of  its  several  forms  to  another  ;  but  it 
cannot  be  created  or  destroyed. 

The  application  of  this  law  to  machines  is  as  follows  :  A  machine 
receives  energy  from  a  source,  and  uses  it  to  do  useful  and  useless 
work.  During  a  complete  cycle  of  action  of  the  machine,  the 
energy  received  equals  the  total  work  done.  In  other  words,  a 
machine  gives  out,  in  some  way,  during  each  cycle,  all  the  energy 
it  receives ;  but  it  cannot  give  out  more  than  it  receives  ;  or,  con- 
sidering a  cycle  of  action, 

energy  received  =  useful  work  +  useless  work. 

When  any  two  of  these  quantities  are  given,  or  can  be  esti- 
mated, the  third  quantity  becomes  known. 


I  MACHINE    DESIGN. 

2.  Function  of  Machines.  —  Nature  furnishes  sources  of  energy, 
and  the  supplying  of  human  needs  requires  ivork  to  be  done.  The 
function  of  machines  is  to  cause  matter  possessing  energy  to  do  useful 
work. 

Illustration. —  The  water  in  a  mill  pond  possesses  energy  (poten- 
tial) by  virtue  of  its  position.  The  earth  exerts  an  attractive  force 
upon  it.  If  there  is  no  outlet,  the  earth's  attractive  force  cannot 
cause  motion  ;  and  hence,  since  motion  is  a  necessary  factor  of 
work,  no  work  is  done. 

If  the  water  overflows  the  dam,  the  earth's  attraction  causes  that 
part  of  it  which  overflows  to  move  to  a  lower  level,  and  before  it  can 
be  brought  to  rest  again,  it  does  work  against  the  force  which  brings 
it  to  rest.  If  this  water  simply  falls  upon  rocks,  its  energy  is 
transformed  into  heat,  with  no  useful  result. 

But  if  the  water  be  led  from  the  pond  to  a  lower  level,  in  a 
closed  pipe  which  connects  with  a  water-wheel,  it  will  exert  pressure 
upon  the  vanes  of  the  wheel  (because  of  the  earth's  attraction),  and 
will  cause  the  wheel  and  its  shaft  to  rotate  against  resistance, 
whereby  it  may  do  useful  work.  The  water-wheel  is  a  machine  and 
is  called  a  Prime  Mover,  because  it  is  the  first  link  in  the  machine 
chain  between  natural  energy  and  useful  work. 

Since  it  is  usually  necessary  to  do  the  required  work  at  some 
distance  from  the  necessary  location  of  the  water-wheel,  Machinery 
of  Transmission  is  used  (shafts,  pulleys,  belts,  cables,  etc.),  and  the 
rotative  energy  is  rendered  available  at  the  required  place. 

But  this  rotative  energy  may  not  be  adapted  to  do  the  required 
work;  the  rotation  may  be  too  slow  or  too  fast;  a  resistance  may 
need  to  be  overcome  in  straight,  parallel  lines,  or  at  periodical  inter- 
vals. Hence  Machinery  of  Application  is  introduced  to  transform 
the  energy  to  meet  the  requirements  of  the  work  to  be  done.  Thus 
the  chain  is  complete,  and  the  potential  energy  of  the  water  does 
the  required  useful  work. 

The  chain  of  machines  which  has  the  steam  boiler  and  engine 
for  its  prime  mover,  transforms  the  potential  heat  energy  of  fuel 
into  useful  work.     This  might  be  analyzed  in  a  similar  way. 


PRELIMINARY.  rt 

3.  Force  Opposed  by  Passive  Resistance.  —  A  force  may  act  without 
being  able  to  produce  motion  (and  hence  without  being  able  to  do 
work),  as  in  the  case  of  the  water  in  a  mill  pond  wdthout  overflow 
or  outlet.  This  may  be  further  illustrated  :  Suppose  a  force,  say 
hand  pressure,  to  be  applied  vertically  to  the  top  of  a  table.  The 
material  of  the  table  offers  a  passive  resistance,  and  the  force  is 
unable  to  produce  motion,  or  to  do  work.  But  if  the  table  top 
were  supported  upon  springs,  the  applied  force  would  overcome  the 
elastic  resistance  of  the  springs,  through  a  certain  space,  and  would 
do  work.  It  is  therefore  possible  to  offer  passive  resistance  to  such 
forces  as  may  be  required  not  to  produce  motion  ;  thereby  rendering 
them  incapable  of  doing  work. 

4.  Constrained  Motion.  —  By  watching  the  action  of  a  machine,  it 
is  seen  that  certain  definite  motions  occur,  and  that  any  departure 
from  these  motions,  or  the  production  of  any  other  motions,  would 
result  in  derangement  of  the  action  of  the  machine.  Thus,  the 
spindle  of  an  engine  lathe  turns  accurately  about  its  axis  ;  the 
cutting  tool  moves  parallel  to  the  spindle's  axis  ;  and  an  accurate 
cylindrical  surface  is  thereby  produced.  If  there  were  any  depart- 
ure from  these  motions,  the  lathe  would  fail  to  do  its  required  work. 
In  all  machines  certain  definite  motions  must  be  produced,  and 
all  other  motions  must  be  prevented  ;  or,  in  other  words,  motion  in 
machines  must  be  constrained. 

In  the  case  of  a  "  free  body  "  acted  on  by  a  system  of  forces,  not 
in  equilibrium,  motion  results  in  the  direction  of  the  resultant  of 
the  system.  If  another  force  be  introduced  whose  line  of  action  does 
not  coincide  with  that  of  the  resultant,  the  direction  of  the  line  of 
action  of  the  resultant  is  changed,  and  the  body  moves  in  this  new 
direction.  The  character  of  the  motion,  therefore,  is  dependent  upon 
the  forces  which  produce  the  motion.     This  is  called  free  motion. 

Example.  —  In  Fig.  1,  suppose  the  free  body  M  to  be  acted  on 
by  the  concurrent  forces  i,  2,  and  3,  whose  lines  of  action  pass 
through  the  center  of  gravity  of  M.  The  line  of  action  of  the  result- 
ant of  these  forces  is  A  B,  and  the  body's  centre  of  gravity  would 
move  along  this  line. 


4  MACHINE    DESIGN, 

If  another  force,  ^,  be  introduced,  CD  becomes  the  line  of  action 
of  the  resultant,  and  the  motion  of  the  body  is  along  the  line  CD. 

Constrained  motion  differs  from  free  motion  in  being  independ- 
ent of  the  forces  which  produce  it.  If  any  force,  not  sufficiently 
great  to  produce  deformation,  be  applied  to  a  body  whose  motion  is 
constrained,  the  result  is  either  a  certain  predetermined  motion,  or 
no  motion  at  all. 

In  a  machine  there  must  be  provision  for  resisting  every  pos- 
sible force  which  tends  to  produce  any  but  the  required  motion. 
This  provision  is  usually  made  by  means  of  the  passive  resistance  of 
properly  formed  and  sufficiently  resistant  metallic  surfaces. 

Illustration  I.  —  Fig.  2  represents  a  section  of  a  wood  lathe 
headstock.  It  is  required  that  the  spindle,  S,  and  the  attached 
cone  pulley,  C,  shall  have  no  other  motion  than  rotation  about  the 
axis  of  the  spindle.  If  any  other  motion  is  possible,  this  machine 
part  cannot  be  used  for  the  required  purpose.  At  A  and  B  the 
cylindrical  surfaces  of  the  spindle  are  enclosed  by  accurately  fitted 
bearings,  or  internal  cylindrical  surfaces.  Suppose  any  force,  P, 
whose  line  of  action  lies  in  the  plane  of  the  paper,  to  be  applied 
to  the  cone  pulley.  It  may  be  resolved  into  a  radial  component, 
R,  and  a  tangential  component,  T.  The  passive  resistance  of  the 
cylindrical  surfaces  of  the  journal  and  its  bearing,  prevents  R  from 
producing  motion  ;  while  it  offers  no  resistance,  friction  being 
disregarded,  to  the  action  of  T,  which  is  allowed  to  produce  the 
required  motion,  i.e.,  rotation  about  the  spindle's  axis.  If  the  line 
of  action  of  P  pass  through  the  axis,  its  tangential  component 
becomes  zero,  and  no  motion  results.  If  the  line  of  action  of 
P  become  tangential,  its  radial  component  becomes  zero,  and 
P  is  wholly  applied  to  produce  rotation.  If  a  force  Q,  whose  line 
of  action  lies  in  the  plane  of  the  paper,  be  applied  to  the  cone, 
it  may  be  resolved  into  a  radial  component,  iV,  and  a,  component,  If, 
parallel  to  the  spindle's  axis.  N  is  resisted  as  before  by  the 
journal  and  bearing  surfaces,  and  M  is  resisted  by  the  shoulder 
surfaces  of  the  bearings,  which  lit  against  the  shoulder  surfaces  of 
the  cone  pulley.    The  force  Q  can  therefore  produce  no  motion  at  all. 


PRELIMINARY.  5 

In  general,  any  force  applied  to  the  cone  pulley  may  be  resolved 
into  a  radial,  a  tangential,  and  an  axial  component.  Of  these  only 
the  tangential  component  is  able  to  produce  motion;  and  that 
motion  is  the  motion  required.  The  constrainment  is  therefore 
complete  ;  i.  e.,  there  can  be  no  motion  except  rotation  about  the 
spindle's  axis.  This  result  is  due  to  the  passive  resistance  of 
metallic  surfaces. 

Illustration  II.  —  i?,  P''ig.  8,  represents,  with  all  details  omitted, 
the  "ram,"  or  portion  of  a  shaping  machine  which  carries  the  cut- 
ting tool.  It  is  required  to  produce  plane  surfaces,  and  hence 
the  "ram"  must  have  accurate  rectilinear  motion  parallel  to  HK. 
Any  deviation  from  such  motion  renders  the  machine  useless. 

Consider  A.  Any  force  which  can  be  applied  to  the  ram,  may 
be  resolved  into  three  components  :  one  vertical,  one  horizontal 
and  parallel  to  the  paper,  and  one  perpendicular  to  the  paper. 
The  vertical  component,  if  acting  upward,  is  resisted  by  the  plane 
surfaces  in  contact  at  C  and  D;  if  acting  downward,  it  is  resisted 
by  the  plane  surfaces  in  contact  at  E.  Therefore  no  vertical  com- 
ponent can  produce  motion.  The  horizontal  component  parallel  to 
the  paper  is  resisted  by  the  plane  surfaces  in  contact  at  F  or  G, 
according  as  it  acts  toward  the  right  or  left.  The  component  per- 
pendicular to  the  paper  is  free  to  produce  motion  parallel  to  its 
line  of  action  ;  but  this  is  the  motion  required. 

Any  force,  therefore,  which  has  a  component  perpendicular  to 
the  paper,  can  produce  the  required  motion;  but  no  other  motion. 
The  constrainment  is  therefore  complete,  and  the  result  is  due  to 
the  passive  resistance  offered  by  metallic  surfaces. 

Complete  Constrainment  is  not  always  required  in  machines.  It 
is  only  necessary  to  prevent  such  motions  as  interfere  with  the 
accomplishment  of  the  desired  result. 

The  weight  of  a  moving  part  is  sometimes  utilized  to  produce 
constrainment  in  one  direction.  Thus  in  a  planer  table,  and  in 
some  lathe  carriages,  downward  motion,  and  unallowable  side 
motion,  are  resisted  by  metallic  surfaces  ;  while  upward  motion  is 
resisted  by  the  weight  of  the  moving  part. 


b  MACHINE    DESIGN. 

Since  the  motions  of  machine  parts  are  independent  of  the  forces 
producing  them,  it  follows  that  the  relation  of  such  motions  may  be 
determined  ivithout  bringing  force  into  the  consideration. 

5.  Kinds  of  Motion  in  Machines.  —  Motion  in  machines  may  be 
very  complex,  but  it  is  chiefly  plane  motion. 

When  a  body  moves  in  such  a  way  that  any  section  of  it 
remains  in  the  same  plane,  its  motion  is  called  plane  motion.  All 
sections  parallel  to  the  above  section  must  also  remain,  each  in  its 
own  plane.  If  the  plane  motion  is  such  that  all  points  of  the 
moving  body  remain  at  a  constant  distance  from  some  line,  AB, 
the  motion  is  called  rotation  about  the  axis  AB.  Example.  —  A 
line  shaft  with  attached  parts. 

If  all  points  of  a  body  move  in  straight  parallel  paths,  the 
motion  of  the  body  is  called  rectilinear  translation.  Examples. — 
Engine  cross-head,  lathe  carriage,  planer  table,  shaper  ram.  Recti- 
linear translation  may  be  conveniently  considered  as  a  special 
case  of  rotation,  in  which  the  axis  of  rotation  is  at  an  infinite 
distance,  at  right  angles  to  the  motion. 

If  a  body  moves  parallel  to  an  axis  about  which  it  rotates,  the 
body  is  said  to  have  helical  or  screw  motion.  Example. —  A  nut 
turning  upon  a  stationary  screw. 

If  all  points  of  a  body,  whose  motion  is  not  plane  motion,  move 
so  that  their  distances  from  a  certain  point,  0,  remain  constant,  the 
motion  is  called  spheric  motion.  This  is  because  the  points  move 
in  the  surface  of  a  sphere  whose  centre  is  0.  Example.  —  The  balls 
of  a  fly-ball  steam-engine  governor,  when  the  position  of  the  valve 
is  changing. 

^6.  Relative  Motion. — The  motion  of  any  machine  part,  like  all 
known  motion,  is  relative  motion.  It  is  studied  by  reference  to 
some  other  part  of  the  same  machine.  Some  one  part  of  a  machine 
is  usually  (though  not  necessarily)  fixed;  i.  e.,  it  has  no  motion 
relatively  to  the  earth.  This  fixed  part  is  called  the  frame  of  the 
machine.  The  motion  of  a  machine  part  may  be  referred  to  the 
frame,  or,  as  often  necessary,  to  some  other  part  which  also  has 
motion  relatively  to  the  frame. 


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PRELIMINARY.  / 

The  kind  and  amount  of  relative  motion  of  a  machine  part, 
depends  upon  tlie  motions  of  the  part  to  which  its  motion  is 
referred. 

RluHtration. —  Fig.  4  shows  a  press.  A  is  the  frame;  C  is  a 
plate  which  is  so  constrained  that  it  may  move  vertically,  but 
cannot  rotate  relatively  to  A.  Motion  of  rotation  is  communicated 
to  the  screw  B.  The  motion  of  B  referred  to  A  is  helical  motion, 
i.  ^.,  combined  rotation  and  translation.  C,  however,  shares  the 
translation  of  i?,  and  hence  there  is  left  only  rotation  as  the  rela- 
tive motion  of  B  and  C.  The  motion  of  B  referred  to  C  is  rotation. 
The  motion  of  C  referred  to  B  is  rotation.  The  motion  of  C 
referred  to  A  is  translation. 

In  general,  if  two  machine  members,  M  and  N,  move  relatively 
to  the  frame,  the  relative  motion  of  M  referred  to  N  depends 
on  how  much  of  the  motion  of  N  is  shared  by  M.  If  M  and  N 
have  the  same  motions  relatively  to  the  frame,  they  have  no  motion 
relatively  to  each  other. 

Conversely,  if  two  bodies  have  no  relative  motion,  they  have  the 
same  motion  relatively  to  a  third  body.  Thus  in  Fig.  4,  if  the  con- 
strainment  of  C  were  such  that  it  could  share  ^'s  rotation^  as  well 
as  its  translation,  then  C  would  have  helical  motion  relatively  to 
the  frame,  and  no  motion  at  all  relatively  to  B.  This  is  assumed 
to  be  self-evident. 

A  rigid  body  is  one  in  which  the  distance  between  elementary 
portions  is  constant.  No  body  is  absolutely  rigid,  but  usually  in 
machine  members  the  departure  from  rigidity  is  so  slight  that  it 
may  be  neglected. 

Many  machine  members,  as  springs,  etc.,  are  useful  because  of 
their  lack  of  rigidity. 

Points  in  a  rigid  body  can  have  no  relative  motion,  and  hence 
■must  all  have  the  same  motion. 

7.  Instantaneous  Motion,  and  Instantaneous  Centres  or  Centros. — 
Points  of  a  moving  body  trace  more  or  less  complex  paths.  If  a 
point  be  considered  as  moving  from  one  position  in  its  path  to 
another  indefinitely  near,  its  motion  is  called  instantaneous  motion. 


8  MACHINE   DESIGN. 

The  point  is  moving,  for  the  instant,  along  a  straight  line  joining 
the  two  indefinitely  near  together  positions,  and  such  a  line  is  a 
tangent  to  the  path.  In  problenris  which  are  solved  by  the  aid  of 
the  conception  of  instantaneous  motion,  it  is  only  necessary  to 
know  the  direction  of  motion  ;  hence,  for  such  purposes,  the  instan- 
taneous motion  of  a  point  is  fully  defined  by  a  tangent  to  its  path 
through  the  point. 

Thus  in  Fig.  5,  if  a  point  is  moving  in  the  path  APB,  when  it 
occupies  the  position  P  the  tangent  TT  represents  its  instantaneous 
motion.  Any  number  of  curves  could  be  drawn  tangent  to  TT  at 
P,  and  any  one  of  them  would  be  a  possible  path  of  the  point ;  but 
whatever  path  it  is  following,  its  instantaneous  motion  is  represented 
by  TT.  The  instantaneous  motion  of  a  point,  is  therefore  independ- 
ent of  the  form  of  its  path.  Any  one  of  the  possible  paths  may  be 
considered  as  equivalent,  for  the  instant,  to  a  circle  whose  centre  is 
anywhere  in  the  normal  NN. 

In  general,  the  instantaneous  motion  of  a  point,  A^  is  equivalent 
to  rotation  about  some  point,  J5,  in  a  line  through  the  point.  A,  per- 
pendicular to  the  direction  of  its  instantaneous  motion. 

Let  the  instantaneous  motion  of  a  point.  A,  Fig.  6,  in  a  section 
of  a  moving  body  be  given  by  the  line  TT.  Then  the  motion  is 
equivalent  to  rotation  about  some  point  of  the  line  AB  as  a  centre  ; 
but  it  may  be  any  point,  and  hence  the  instantaneous  motion  of  the 
body  is  not  determined.  But  if  the  instantaneous  motion  of 
another  point,  C,  be  given  by  the  line  TiT^,  this  motion  is  equiva- 
lent to  rotation  about  some  point  of  CD.  But  the  points  A  and  C 
are  points  in  a  rigid  body,  and  can  have  no  relative  motion,  and 
must  have  the  same  motion,  i.  e.,  rotation  about  the  same  centre. 
A  rotates  about  some  point  of  AB,  and  C  rotates  about  some  point 
of  CD;  but  they  must  rotate  about  the  same  point,  and  the  only 
point  which  is  at  the  same  time  in  both  lines,  is  their  intersection,  0. 
Hence  A  and  O,  and  all  other  points  of  the  body,  rotate  for  the 
instant  about  an  axis  of  which  0  is  the  projection  ;  or,  in  other 
words,  the  instantaneous  motion  of  the  body  is  rotation  about  an 
axis  of  which  0  is  the  projection.     This  axis  is  the  instantaneous 


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PRELIMINARY.  U 

axis  of  the  body's  motion,  and  0  is  the  instantaneous  centre  of  the 
motion  of  the  section  shown  in  Fig.  6. 

For  the  sake  of  brevity  an  instantaneous  centre  will  be  called  a 
centre. 

If  TT  and  T,T,  had  been  parallel  to  each  other,  AB  and  CD 
would  also  have  been  parallel,  and  would  have  intersected  at  in- 
finity ;  in  which  case  the  body's  instantaneous  motion  would  have 
been  rotation  about  an  axis  infinitely  distant  ;  i.  e.,  it  would  have 
been  translation. 

The  motion  of  the  body  in  Fig.  6,  is  of  course  referred  to  a  fixed 
body,  which,  in  this  case,  may  be  represented  by  the  paper.  The 
instantaneous  motion  of  the  body  is  rotation  about  0  relatively 
to  the  paper.  Let  M  represent  the  figure,  and  N  the  fixed  body 
represented  by  the  paper.  Suppose  the  material  of  M  to  be  ex- 
tended so  as  to  include  0.  Then  a  pin  could  be  put  through  0, 
materially  connecting  M  and  N,  without  interfering  with  their 
instantaneous  motion.  Such  connection  at  any  other  point  would 
interfere  with  the  instantaneous  motion. 

The  centra  of  the  relative  motion  of  two  bodies  is  a  point,  and  the 
only  one,  at  which  they  have  no  relative  motion ;  it  is  a  point,  and  the 
only  one,  that  is  common  to  the  two  bodies  for  the  instant. 

It  will  be  seen  that  the  points  of  the  figure  in  Fig.  6  might  be 
moving  in  any  paths,  so  long  as  those  paths  are  tangent  at  the 
points  to  the  lines  representing  the  instantaneous  motion. 

In  general,  centros  of  the  relative  motion  of  two  bodies  are  con- 
tinually changing  their  position.  They  may,  however,  remain  sta- 
tionary; i.  e.,  they  may  become  fixed  centres  of  rotation. 

8.  Loci  of  Centros,  or  Centroids.  —  As  centros  change  position  they 
describe  curves  of  some  kind,  and  these  loci  of  centros  may  be 
called  centroids. 

Suppose  a  section  of  any  body,  M,  to  have  motion  relatively  to  a 
section  of  another  body,  jV  (fixed),  in  the  same  or  parallel  plane. 
Centros  may  be  found  for  a  series  of  positions,  and  a  curve  drawn 
through  them  on  the  plane  of  N  would  be  the  centroid  of  the  motion 
of  M  relatively  to  N.     If,  now,  M  being  fixed,  N  moves  so  that  the 


10  MACHINE    DESIGN. 

relative  motion  is  the  same  as  before,  the  centroid  of  the  motion  of 
N  relatively  to  M,  may  be  located  upon  the  plane  of  M.  Now,  since 
the  centro  of  the  relative  motion  of  two  bodies  is  a  point  at  which 
they  have  no  relative  motion,  and  since  the  points  of  the  centroids 
become  successively  the  centros  of  the  relative  motion,  it  follows 
that  as  the  motion  goes  on,  the  centroids  would  roll  upon  each 
other  without  slipping.  Therefore,  if  the  centroids  are  drawn,  and 
rolled  upon  each  other  without  slipping,  the  bodies  M  and  N  will 
have  the  same  relative  motion  as  before.  From  this  it  follows  that 
the  relative  plane  motion  of  two  bodies  may  be  reproduced  by  roll- 
ing together,  without  slipping,  the  centroids  of  that  motion. 

9.  Pairs  of  Motion  Elements. — ^The  external  and  internal  surfaces 
by  which  motion  is  constrained  in  Figs.  2  and  8  may  be  called 
pairs  of  motion  elements.  The  pair  in  Fig.  2  is  called  a  turning  pair, 
and  the  pair  in  Fig.  3  is  called  a  sliding  pair. 

The  helical  surfaces  by  which  a  nut  and  screw  engage  with  each 
other,  are  called  a  twisting  pair.  These  three  pairs  of  motion  ele- 
ments have  their  surfaces  in  contact  throughout.  They  are  called 
lower  pairs.  Another  class,  called  higher  pairs,  have  contact  only 
along  elements  of  their  surfaces.  Examples. —  Cams  and  toothed 
wheels. 


CHAPTER   11. 

MOTION    IN    MECHANISMS. 

10.  Linkages  or  Motion  Chains;  Mechanisms. 

In  Fig.  7,  h  is  joined  to  c  by  a  turning  pair 
c  "    ^        d     "    sliding       " 

rf  "  a     "    turning      " 

a  "  ^     u 

Evidently  there  is  complete  constrainment  of  the  relative  motion 
of  a,  6,  c,  and  d.  For,  d  being  fixed,  if  any  motion  occurs  in  either 
a,  b,  or  c,  the  other  two  must  have  a  predetermined  corresponding 
motion. 

c  may  represent  the  cross-head,  b  the  connecting-rod,  and  a  the 
crank  of  a  steam  engine  of  the  ordinary  type.  If  r  were  rigidly 
attached  to  a  piston  upon  which  the  expansive  force  of  steam 
acts  toward,  the  right,  a  must  rotate  about  ad.  This  represents  a 
m.achinfi.  The  members  a,  b,  c,  and  d,  may  be  represented  for  the 
study  of  relative  motions  by  the  diagram,  Fig.  8. 

This  assemblage  of  bodies,  connected  so  that  there  is  complete 
constrainment  of  motion,  may  be  called  a  motion  chain  or  linkage, 
and  the  connected  bodies  may  be  called  links.  The  chain  shown  is 
a  simple  chain,  because  no  link  is  joined  to  more  than  two  others. 
If  any  of  the  links  of  a  chain  are  joined  to  more  than  two 
others,  the  chain  is  a  compound  chain.     Examples  will  be  given  later. 

When  one  link  of  a  chain  is  fixed,  i.  e.,  when  it  becomes  the 
standard  to  which  the  motion  of  the  others  is  referred,  the  chain  is 
called  a  mechanism.    Fixing  different  links  of  a  chain  gives  differ- 


12  MAGHINK    DESIGN. 

ent  mechanisms.  Thus  in  Fig.  8,  if  d  be  fixed,  the  mechanism  is 
that  which  is  used  in  the  usual  type  of  steam  engine,  as  in  Fig.  7. 
It  is  called  the   slider  crank  mechanism. 

But  if  a  be  fixed,  the  result  is  an  entirely  different  mechanism  ; 
for  h  would  then  rotate  about  the  permanent  centre  ah,  d  would 
rotate  about  the  permanent  centre  ad,  while  c  would  have  a  more 
complex  motion,  rotating  about  a  constantly  changing  centro, 
whose  path  may  be  found. 

Fixing  h  or  c  would  give,  in  each  case,  a  different  mechanism. 

11.  Location  of  Centros.  —  In  Fig.  8  d  is  fixed  and  it  is  required 
to  find  the  centros  of  rotation,  either  permanent  or  instantaneous, 
of  the  other  three  links.  The  motion  of  a,  relatively  to  the  fixed 
link  d,  is  rotation  about  the  fixed  centre  ad.  The  motion  of  r. 
relatively  to  d  is  translation,  or  rotation  about  a  centro  cd,  at  infin- 
ity vertically.  The  link  b  has  a  point  in  common  with  a  ;  it  is 
the  centro,  ah,  of  their  relative  motion.  This  point  may  be  con- 
sidered as  a  point  in  a  or  6  ;  in  either  case  it  can  have  but  one 
direction  of  motion.  As  a  point  in  a  its  motion,  relatively  to  d, 
is  rotation  about  ad.  For  the  instant,  then,  it  is  moving  along  a 
tangent  to  the  circle  through  ah.  But  as  a  point  in  h,  its  direction 
of  instantaneous  motion  must  be  the  same,  and  hence  its  motion 
must  be  about  some  point  in  the  line  ad^ah,  extended  if  necessary. 
Also  h  has  a  point,  he,  in  common  with  c  ;  and  by  the  same 
reasoning  as  above,  he,  as  a  point  in  h,  rotates,  for  the  instant, 
about  some  point  of  the  vertical  line  through  he.  Now  ah  and  he 
are  points  of  a  rigid  body,  and  one  rotates  for  the  instant  about 
some  point  of  AB ;  and  the  other  rotates  for  the  instant  about 
some  point,  CD]  hence  both  (as  well  as  all  other  points  of  6) 
must  rotate  about  the  intersection  of  AB  and  CD.  Hence  hd  is 
the  centro  of  the  motion  of  h  relatively  to  d. 

The  motion  of  a  may  be  referred  to  e  (fixed),  and  ac  will  be 
found  (by  reasoning  like  that  applied  to  h)  to  lie  at  the  intersec- 
tion of  the  lines  EF  and  GH. 

The  motion  chain  in  Fig.  8,  as  before  stated,  is  called  the  slider 
crank  chain. 


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MOTION    IN    MECHANISMS.  18 

12.  Centres  of  the  Relative  Motion  of  Three  Bodies  Are  Always  in 
the  Same  Straight  Line.  —  In  Fig.  8  it  will  be  seen  that  the  three 
centres  of  any  three  links  lie  in  the  same  straight  line.  Thus  ad, 
ah,  and  hd,  are  the  centres  of  the  links  a,  h,  and  d.  This  is  true  of 
any  other  set  of  three  links.  Proof. —  Consider  a,  6,  and  c.  The  centro 
ah  as  a  point  in  a  has  a  direction  of  instantaneous  motion  perpen- 
dicular to  a  line  joining  it  to  ad.  As  a  point  in  h  it  has  a  direction 
of  instantaneous  motion  perpendicular  to  a  line  joining  it  to  hd. 
Therefore  the  lines  ah — ad  and  ah  —  hd  are  both  perpendicular  to 
the  direction  of  instantaneous  motion  of  a?),  and  they  also  both  pass 
through  ah  ;  hence  they  must  coincide,  and  therefore  ah,  ad,  and  hd 
must  lie  in  the  same  straight  line.  But  a,  h,  and  d  might  be  any 
three  bodies  whatever,  which  have  relative  plane  motion,  and  the 
above  reasoning  would  hold.  Hence  it  may  be  stated  :  The  three 
centros  of  any  three  hodies  having  relative  plane  motion,  must  lie  in 
the  same  straight  line.  [The  statement  and  proof  of  this  important 
proposition  is  due  to  Prof.  Kennedy.] 

13.  Lever  Crank  Chain.  Location  of  Centros.  —  Fig.  9  shows  a 
chain  of  four  links  of  unequal  length  joined  to  each  other  by  turn- 
ing pairs.  The  centros  ah,  ad,  cd,  and  he  may  be  located  at  once, 
since  they  are  the  centros  of  turning  pairs  which  join  adjacent 
links  to  each  other.  The  centros  of  the  relative  motion  of  h,  c,  and 
d  are  he,  cd,  and  hd,  and  these  must  be  in  the  same  straight  line. 
Hence  hd  is  in  the  line  B.  The  centros  of  the  relative  motion  of 
a,  h,  and  d,  are  ah,  hd,  and  ad  ;  and  these  also  must  lie  in  the  same 
straight  line.  Hence  hd  is  in  the  line  A.  Being  at  the  same  time 
in  A  and  B.  it  must  be  at  their  intersection. 

14.  The  Constrainment  of  Motion  in  a  linkage  is  independent  of 
the  size  of  the  motion  elements.  As  long  as  the  cylindrical  surfaces 
of  turning  pairs  have  their  axes  unchanged,  the  surfaces  themselves 
may  be  of  any  size  whatever,  and  the  motion  is  unchanged.  The 
same  is  true  of  sliding  and  twisting  pairs. 

In  Fig.  10,  suppose  the  turning  pair  connecting  c  and  d  to  be 
enlarged  so  that  it  includes  he.  The  link  e  now  becomes  a  cylinder, 
turning  in  a  ring  attached  to,  and  forming  part  of,  the  link  h.     he 


14  MACHINE    DESIGN. 

becomes  a  pin  made  fast  in  c  and  engaging  with  an  eye  at  the  end 
of  h.  The  centros  are  the  same  as  before  the  enlargement  of  cd^  and 
hence  the  relative  motion  is  the  same. 

In  Fig.  11,  the  circular  portion  immediately  surrounding  cd  is 
attached  to  d.  The  link  c  now  becomes  a  ring  moving  in  a  circular 
slot.  This  may  be  simplified  as  in  Fig.  12,  whence  c  becomes  a 
curved  block  moving  in  a  limited  circular  slot  in  d.  The  centros 
remain  as  before,  the  relative  motion  is  the  same,  and  the  linkage 
is  essentially  unchanged. 

15.  If,  in  the  slider  crank  mechanism,  the  turning  pair  whose 
axis  is  a6,  be  enlarged  till  ad  is  included,  as  in  Fig.  18,  the  motion  of 
the  mechanism  is  unchanged,  but  the  link  a  is  now  called  an  eccen- 
tric instead  of  a  crank.  This  mechanism  is  usually  used  to  com- 
municate motion  from  the  main  shaft  of  a  steam  engine  to  the 
valve.  It  is  used  because  it  may  be  put  on  the  main  shaft  any- 
where, without  interfering  with  its  continuity  and  strength. 

The  mechanism  shown  in  Fig.  14  is  called  the  "  slotted  cross- 
head  mechanism. ^^  Its  centros  may  be  found  from  principles 
already  given. 

This  mechanism  is  often  used  as  follows  :  One  end  of  c,  as  E,  is 
attached  to  a  piston  working  in  a  cylinder  attached  to  d.  This 
piston  is  caused  to  reciprocate  by  the  expansive  force  of  steam  or 
some  other  fluid.  The  other  end  of  c  is  attached  to  another  piston, 
which  also  works  in  a  cylinder  attached  to  d.  This  piston  may 
pump  water,  or  compress  gas.  The  crank  a  is  attached  to  a  shaft, 
the  projection  of  whose  axis  is  ad.  This  shaft  also  carries  a  fly- 
wheel which  insures  approximately  uniform  rotation. 

16.  Location  of  Centros  in  a  Compound  Mechanism. — It  is  required 
to  find  the  centros  of  the  compound  linkage.  Fig.  15.  In  any  link- 
age, each  link  has  a  centro  relatively  to  every  other  link  ;  hence,  if 
the  number  of  links  =  n,  the  number  of  centros^:n(n  —  1).  But 
the  centro  ah  is  the  same  as  ba  ;  i.  e.,  each  centro  is  double.     Hence 

the  number  of  centros  to  be  located  for  any  linkage  =  —^^ — ^—  .    In 

6x5 
the  linkage  Fig.  15,  the  number  of  centros  =  — ^ —  =  15. 


s- 


^^^^^ 


or  THB 


^ 


TJSITBESIT 


OS* 


ilPO^ 


MOTION    IN    MECHANISMS.  15 

The  portion  above  the  link  d  is  a  slider  crank  chain,  and  the 
character  of  its  motion  is  in  no  way  affected  by  the  attachment  of 
the  part  below  d.  On  the  other  hand,  the  lower  part  is  a  lever 
crank  chain,  and  the  character  of  its  motion  is  not  affected  by  its 
attachment  to  the  upper  part.  The  chain  may  therefore  be  treated 
in  two  parts,  and  the  centros  of  each  part  may  be  located  from 
what  has  preceded.  Each  part  will  have  six  centros,  and  twelve 
would  thus  be  located,  ad,  however,  is  common  to  the  two  parts,  and 
hence  only  eleven  are  really  found.  Four  centros,  therefore,  remain 
to  be  located.  They  are  be,  cf,  hf,  and  ce.  To  locate  be,  consider 
the  three  links  a,  b,  and  e,  and  it  follows  that  be  is  in  the  line  A 
passing  through  ab  and  ae  ;  considering  b,  d,  and  e,  it  follows  that 
be  is  in  the  line  B  through  bd  and  de.  Hence  be  is  at  the  intersec- 
tion of  A  and  B.     Similar  methods  locate  the  other  centros. 

In  general,  for  finding  the  centros  of  a  compound  linkage  of  six 
links,  consider  the  linkage  to  be  made  up  of  two  simple  chains,  and 
find  their  centros  independently  of  each  other.  Then  take  the  two 
links  whose  centro  is  required,  together  with  one  of  the  links  carry- 
ing three  motion  elements  (as  a.  Fig.  15).  The  centros  of  these 
links  locate  a  straight  line.  A,  which  contains  the  required  centro. 
Then  take  the  two  links  whose  centro  is  required,  together  with  the 
other  link  which  carries  three  motion  elements.  A  straight  line,  B, 
is  thereby  located,  which  contains  the  required  centro,  and  the 
latter  is  therefore  at  the  intersection  of  A  and  B. 

17.  Velocity  is  rate  of  motion,  or  motion  per  unit  time. 

Linear  velocity  is  linear  space  moved  through  in  unit  time  ;  it 
may  be  expressed  in  any  units  of  length  and  time  ;  as  miles  per 
hour,  feet  per  minnte  or  per  second,  etc. 

Angular  velocity  is  angular  space  moved  through  in  unit  time. 
In  machines,  angular  velocity  is  usually  expressed  in  revolutions 
per  minute  or  per  second. 

The  linear  space  described  by  a  point  in  a  rotating  body,  or  its 
linear  velocity,  is  directly  proportional  lo  its  radius,  or  its  distance 
from  the  axis  of  rotation.  This  is  true  because  arcs  are  propor- 
tional to  radii. 


16  MACHINE    DESIGN. 

If  A  and  B  are  two  points  in  a  rotating  body,  and  if  Vy  and  r.^ 
are  their  radii,  then  the  ratio  of  linear  velocities 

linear  veloc.  A      r^ 

linear  veloc.  B      r.^  ' 

This  is  true  whether  the  rotation  is  about  a  centre  or  a  centro  ; 
i.  <?.^it  is  true  both  for  continuous  or  instantaneous  rotation.  Hence 
it  applies  to  all  cases  of  plane  motion  in  machines  ;  because  all 
plane  motion  in  machines  is  equivalent  to  either  continuous  or 
instantaneous  rotation  about  some  point. 

To  find  the  relation  of  linear  velocity  of  two  points  in  a  machine 
member,  therefore,  it  is  only  necessary  to  find  the  relation  of  the 
radii  of  the  points.  The  latter  relation  can  easily  be  found  when 
the  centre  or  centro  is  located. 

18.  A  vector  is  something  which  has  magnitude  and  direction. 
A  vector  may  be  represented  by  a  straight  line,  because  the  latter 
has  magnitude  (its  length)  and  direction.  Thus  the  length  of  a 
straight  line,  AB,  may  represent,  upon  some  scale,  the  magnitude  of 
some  vector,  and  it  may  represent  the  vector's  direction  by  being 
parallel  to  it,  or  by  being  perpendicular  to  it.  For  convenience 
the  latter  plan  will  here  be  used.  The  vectors  to  be  represented  are 
the  linear  velocities  of  points  in  mechanisms.  The  lines  which 
represent  vectors  are  also  called  vectors. 

A  line  which  represents  the  linear  velocity  of  a  point,  will  be 
called  the  linear  velocity  vector  of  the  point.  The  symbol  of  linear 
velocity  will  be  VI.  Thus  VIA  is  the  linear  velocity  of  the  point 
A.     Also  Va  will  be  used  as  the  symbol  of  angular  velocity. 

If  the  linear  velocity  and  radius  of  a  point  are  known,  the 
angular  velocity,  or  the  number  of  revolutions  per  unit  time,  may 
be  found  ;  since  the  linear  velocity  h-  length  of  the  circumference  in 
which  the  point  travels  =  angular  velocity. 

All  points  of  a  rigid  body  have  the  same  angular  velocity. 

If  the  radii,  and  ratio  of  linear  velocities  of  two  points,  in 
different  machine  members,  are  known,  the  ratio  of  the  angular 
velocities  of  the  members  may  be  found  as  follows  : 

Let  ^  be  a  point  in  a  member  M,  and  B  a  point  in  a  member  N. 


VaM 


MOTION    IN    MECHANISMS.  17 

/•i  =  radius  of  A;  r.,  =  radius  of  B.      VIA  and  VIB  represent  the 

VIA 

linear  velocities  of  A  and  B,  whose  ratio,  777^,  is  known. 

V  IB 

Then  VaA  =  ^^    and    VaB  =  ^. 

VaA  _  VIA       '2r:r,       VIA  ^^  r.,  _  VaM 
""^  VaB~  2^r,  ^  VIB~  VIB  ^  r,  "  VaN' 

If  M  and  N  rotate  uniformly  about  fixed  centres,  the  ratio 

VaN 

is  constant.  If  either  M  or  N  rotates  about  a  centro,  the  ratio  is  a 
varying  one. 

19.  To  find  the  relation  of  linear  velocity  of  two  points  in  the 
same  link,  it  is  only  necessary  to  measure  the  radii  of  the  points, 
and  the  ratio  of  these  radii  is  the  ratio  of  the  linear  velocities  of 
the  points. 

In  Fig.  16,  let  the  smaller  circle  represent  the  path  of  A,  the 

centre  of  the  crank  pin  of  a  slider  crank  mechanism  ;  the  link  d 

being  fixed.     Let  the  larger  circle  represent  the  rim  of  a  pulley, 

which  is  keyed  to  the  same  shaft  as  the  crank.     The  pulley  and  the 

crank  are  then  parts  of  the  same  link.     The  ratio  of  velocity  of  the 

VIA        r 
crank  pin  centre  and  the  pulley  surface  =  ^tTTn  ^=~  •     ^^^  ^^is  case 

the  link  rotates  about  a  fixed  centre.  The  same  relation  holds, 
however,  when  the  link  rotates  about  a  centro. 

20.  In  Fig.  17,  the  link  d  is  fixed  and  -777-1-  =  -, :rT . 

VI  he       he  —  hd 

By  similar  triangles  this  expression  is  also  equal  to  jz ^  • 

Hence,  if  the  radius  of  the  crank  circle  be  taken  as  the  vector  of 
the  constant  linear  velocity  of  a6,  the  distance  cut  off  on  the  verti- 
cal through  0  by  the  line  of  the  connecting-rod  (extended  if 
necessary)  will  be  the  vector  of  the  linear  velocity  of  he.  Project 
A    horizontally    upon    />c  —  hd,    locating   B.     Then   he  —  B   is   the 


18  MACHINE    DESIGN. 

vector  of  VI  of  the  slider,  and  may  be  used  as  an  ordinate  of  the 
linear  velocity  diagram  of  the  slider.  By  repeating  the  above 
construction  for  a  series  of  positions,  the  ordinates  representing  the 
VI  of  he  for  different  positions  of  the  slider  may  be  found.  A 
smooth  curve  through  the  extremities  of  these  ordinates  is  the 
velocity  curve,  from  which  the  Vh  for  all  points  of  the  slider's 
stroke  may  be  read.  The  scale  of  velocities,  or  the  linear  velocity 
represented  by  one  inch  of  ordinate,  equals  the  constant  linear 
velocity  of  ab  divided  by  0  —  ah  in  inches. 

21.  It  is  required  to  find  VI  of  be  during  a  cycle  of  action  of  the 
mechanism  shown  in  Fig.  18,  d  being  fixed,  and  VI  of  ah  being  con- 
stant. The  two  points,  ah  and  he,  may  both  be  considered  in  the 
link  h.     All  points  in  b  move  about  bd  relatively  to  the  fixed  link. 

-r_.  Vlab      ab  —  bd 

But  a  line,  as  MN,  drawn  parallel  to  h  cuts  off  on  the  radii  portions 
which  are  proportion tal  to  the  radii  themselves,  and  hence  propor- 
tional to  the  Vis  of  the  points.     Hence 

Vlab      ah  —  M 


Vibe       he  —  N' 


The  arc  in  which  he  moves  may  be  divided  into  any  number  of 
parts,  and  the  corresponding  positions  of  ab  may  be  located.  A 
circle  through  M.  about  ad,  may  be  drawn,  and  the  constant  radial 
distance  ah  —  M  may  represent  the  constant  velocity  of  ab. 
Through  M„  M.^,  etc.,  draw  lines  parallel  to  the  corresponding 
positions  of  b,  and  these  lines  will  cut  off  on  the  corresponding  line 
of  c  a  distance  which  represents  VI  of  he.  Through  the  points  thus 
determined  the  velocity  diagram  may  be  drawn,  and  the  VI  of  he 
for  a  complete  cycle  is  determined.  The  scale  of  velocities  is  found 
as  in  §  20. 

22.   The  relation  of  linear  velocity  of  points   not  in   the  same 
link  may  also  be  found. 


^^t. 


MOTION    IN    MECHANISMS.  19 

Required   „  .      The  centro  ah  is  a  point  common  to  a  and 

h,  the  two  links  considered  ;  d  is  the  fixed  link.  Consider  ab  as  a 
point  in  a  ;  and  its  VI  is  to  that  of  A  as  their  radii  or  distances  from 
ad.  Draw  a  vector  triangle  with  its  sides  parallel  to  the  triangle 
formed  by  joining  A,  ah,  and  ad.  Then  if  the  side  A^  represent  the 
VI  of  A,  the  side  ah^  will  represent  the  VI  of  ah.  Consider  ah  as  a 
point  in  h,  and  its  VI  is  to  that  of  B  as  their  radii,  or  distances  to 
hd.  Upon  the  vector  ah^,  draw  a  triangle  whose  sides  are  parallel 
to  those  of  a  triangle  formed  by  joining  ah,  hd,  and  B.  Then,  from 
similar  triangles,  the  side  B^  is  the  vector  of  ^'s  linear  velocity. 

VI  oi  A vector  A^ 

^^"^®  VlofB^  vector  B, ' 

The  path  of  B  during  a  complete  cycle  may  be  traced,  and  the 
VI  for  a  series  of  points  may  be  found,  by  the  above  method  ;  then 
the  vectors  may  be  laid  off  on  normals  to  the  path  through  the 
points  ;  the  velocity  curve  may  be  drawn  ;  and  the  velocity  of  B  at 
all  points  becomes  known. 

23.  The  diagram  of  VI  of  the  slider  of  the  slider  crank  mechan- 
ism, Fig.  17,  is  unsymmetrical  with  respect  to  a  vertical  axis  through 
its  centre.  This  is  due  to  the  angularity  of  the  connecting-rod,  and 
may  be  explained  as  follows  : 

In  Fig.  20,  A  is  one  angular  position  of  the  crank,  and  B  is  the 
corresponding  angular  position  on  the  other  side  of  the  vertical 
through  the  centre  of  rotation.  The  corresponding  positions  of  the 
slider  are  as  shown.  But  for  position  ^,the  line  of  the  connecting- 
rod,  C,  cuts  off  on  the  vertical  through  0,  a  vector  Oa,  which 
represents  the  slider's  velocity.  For  position  B  the  vector  of  the 
slider's  velocity  is  Oh.  Obviously  this  difference  is  due  to  the 
angularity  of  the  connecting-rod. 

In  a  mechanism  which  is  equivalent  to  the  slider  crank  with 
the  connecting-rod  always  horizontal  (as  the  slotted  cross-head) 
the  line  of  the  connecting-rod  would  cut  off  on  OF  the  same  vector 


20  MACHINE    DESIGN. 

for  position  A  and  position  B.  Hence  the  velocity  diagram  for  the 
slotted  cross-head  mechanism  is  symmetrical  with  respect  to  both 
vertical  and  horizontal  axes  through  its  centre.  In  fact,  if  the 
crank  radius  (=  length  of  link  (t)  be  taken  as  the  vector  of  the  VI 
of  ab,  the  linear  velocity  diagram  of  the  slider  becomes  a  circle 
whose  radius  =:  the  length  of  the  link  a.  Hence  the  crank  circle 
itself  serves  for  the  linear  velocity  diagram,  the  horizontal  diameter 
representing  the  path  of  the  slider. 

24.  During  a  portion  of  the  cycle  of  the  slider  crank  mechanism, 
the  slider's  VI  is  greater  than  that  of  ab.  This  is  also  due  to  the 
angularity  of  the  connecting-rod,  and  may  be  explained  as  follows  : 
In  Fig.  21,  as  the  crank  moves  up  from  the  position  x,  it  will  reach 
such  a  position.  A,  that  the  line  of  the  connecting-rod  extended  will 
pass  through  B.  OB  in  this  position  is  the  vector  of  the  linear 
velocity  of  both  ab  and  the  slider,  and  hence  their  linear  velocities 
are  equal.  When  ab  reaches  B,  the  line  of  the  connecting-rod  passes 
through  B;  and  again  the  vectors  —  and  hence  the  linear  velocities  — 
of  ab  and  the  slider  are  equal.  For  all  positions  between  A  and  B. 
the  line  of  the  connecting-rod  will  cut  OB  outside  of  the  crank 
circle ;  and  hence  the  linear  velocity  of  the  slider  will  be  greater 
than  that  of  ab.  Obviously,  the  linear  velocity  of  the  slider  is. 
greatest  when  the  angle  between  crank  and  connecting-rod  ^=  90°. 
This  result  is  due  to  the  angularity  of  the  connecting-rod,  because 
if  the  latter  remained  always  horizontal,  its  line  could  never  cut  OB 
outside  the  circle.  It  follows  that  in  the  slotted  cross-head  mechan- 
ism the  maximum  VI  of  the  slider  =^  the  constant  VI  of  ab.  The 
angular  space  BOA,  Fig.  21,  throughout  which  VI  of  the  slider  is 
greater  than  the  VI  of  ab,  increases  with  increase  of  angularity  of 
the  connecting-rod  ;  /.  e.,  it  increases  with  the  ratio 

length  of  crank 
length  of  connecting-rod* 

25.  A  slider  in  a  mechanism  often  carries  a  cutting  tool,  which 
cuts  during  its  motion  in  one  direction,  and  is  idle  during  the  return 


'■K,^^   or  THJI     *^ 

UKIVBRSI 


MOTION    IN    MECHANISMS.  21 

stroke.  Sometimes  the  slider  carries  the  piece  to  be  cut,  and  the 
cutting  occurs  while  it  passes  under  a  tool  made  fast  to  the  fixed 
link,  the  return  stroke  being  idle. 

The  velocity  of  cutting  is  limited.  If  the  limiting  velocity  be 
exceeded,  the  tool  becomes  so  hot  that  its  temper  is  drawn,  and  it 
becomes  unfit  for  cutting.  The  limit  of  cutting  velocity  depends  on 
the  nature  of  the  material  to  be  cut.  Thus  annealed  tool-steel,  and 
the  scale  surface  of  cast  iron,  may  be  cut  at  20  feet  per  minute  ; 
wrought  iron  and  soft  steel  at  25  to  30  feet  per  minute  ;  while 
brass  and  the  softer  alloys  may  be  cut  at  40  or  more  feet  per 
minute.  There  is  no  limit  of  this  kind,  however,  to  the  velocity 
during  the  idle  stroke  ;  and  it  is  desirable  to  make  it  as  great  as 
possible^  in  order  to  increase  the  product  of  the  machine.  This 
leads  to  the  design  and  use  of  "quick  return"  mechanisms. 

26.  If,  in  a  slider  crank  mechanism,  the  centre  of  rotation  of  the 
crank  be  moved,  so  that  the  line  of  the  slider's  motion  does  not  pass 
through  it,  the  slider  will  have  a  quick  return  motion. 

In  Fig.  22,  when  the  slider  is  in  its  extreme  position  at  the 
right,  the  crank-pin  centre  is  at  D.  When  the  slider  is  at  B,  the 
crank-pin  centre  is  at  C.  If  rotation  is  as  indicated  by  the  arrow, 
then  while  the  slider  moves  from  B  to  A,  the  crank-pin  centre 
moves  from  C  over  to  D.  And  while  the  slider  returns  from  A  to 
B,  the  crank-pin  centre  moves  under  from  D  to  C.  If  the  VI  of  the 
crank-pin  centre  be  assumed  constant,  the  time  occupied  in  moving 
from  /)  to  C  is  less  than  that  from  C  to  D.  Hence,  the  time  occu- 
pied by  the  slider  in  moving  from  B  to  A  is  greater  than  that 
occupied  in  moving  from  A  to  B.  The  mean  velocity  during  the 
forward  stroke  is  therefore  less  than  during  the  return  stroke.  Or 
the  slider  has  a  "quick  return"  motion. 

It  is  required  to  design  a  mechanism  of  this  kind  for  a  length  of 
stroke  =  BA  and  for  a  ratio 

mean  VI  forward  stroke 5 

mean  VI  return  stroke       7 

The  mean  velocity  of  either  stroke  is  inversely  proportional  to  the 


22  MACHINE    DESIGN. 

time  occupied,  and  the  time  is  proportional  to  the  corresponding 
angle  described  by  the  crank.     Hence 

mean  velocity  forward  _  5 angle  /5 

mean  velocity  return        7       angle  « * 

It  is  therefore  necessary  to  divide  360°  into  two  parts  which  are  to 
each  other  as  5  to  7.  Hence  a  =^210"  and  1^=150°.  Obviously 
ff=zlSO° — i3^'d0°.  Place  the  30°  angle  of  a  drawing  triangle  so 
that  its  sides  pass  through  B  and  A.  This  condition  may  be  ful- 
filled and  yet  the  vertex  of  the  triangle  may  occupy  an  indefinite 
number  of  positions.  By  trial  0  may  be  located  so  that  the  crank 
shall  not  interfere  with  the  line  of  the  slider.  0  being  located 
tentatively,  it  is  necessary  to  find  the  corresponding  lengths  of 
crank  a,  and  connecting-rod  h.  When  the  crank-pin  centre  is  at 
/),  A0=  b  —  a  ;  when  it  is  at  C,  BO  =  h  -j-  a.  AO  and  BO  are  meas- 
urable values  of  length  ;  hence  a  and  h  may  be  found,  the  crank 
circle  may  be  drawn,  and  the  velocity  diagrams  may  be  constructed 
as  in  Fig.  17;  remembering  that  the  distance  cut  off  upon  a  vertical 
through  0,  by  the  line  of  the  connecting-rod,  is  the  vector  of  the  VI 
of  the  slider  for  the  corresponding  position,  when  the  VI  of  the  crank- 
pin  centre  is  represented  by  the  crank  radius. 

The  construction  may  be  checked  by  finding  the  mean  heights  of 
the  velocity  diagram,  or  the  areas,  which  are  proportional  to  the 
mean  heights,  above  and  below  the  horizontal  line  ;  these  should 
be  to  each  other  as  5:7.  The  areas  may  be  found  by  use  of  a 
planimeter  ;  and  these  areas,  divided  by  the  length  of  stroke,  equal 
the  mean  heights. 

It  is  required  to  make  the  maximum  velocity  of  the  forward 
stroke  of  the  slider  ^  20  feet  per  minute,  and  to  find  the  correspond- 
ing number  of  revolutions  per  minute  of  the  crank.  The  maximum 
linear  velocity  vector  of  the  forward  stroke^  the  maximum  height 
of  the  upper  part  of  the  velocity  diagram  ;  call  it  V^.  Call  the 
linear  velocity  vector  of  the  crank-pin  centre  V.^  =  crank  radius. 
Let  x^  linear  velocity  of  the  crank-pin  centre.     Then 


^-^A^   Of  THE        ^ 

:uiriyERSiT7: 


MOTION    IN    MECHANISMS.  23 

l\  _  20  ft.  per  minute 


X 


20  ft.  per  minute  X  V., 

or  x  = ^ Y7 • 

'^  1 

a-  is  therefore  expressed  in  known  terms.  If  now  x,  the  space  the 
crank-pin  centre  is  required  to  move  through  per  minute,  be  divided 
by  the  space  moved  through  per  revolution,  the  result  will  equal  the 
number  of  revolutions  per  minute  =  N ; 

N- 


2?:  X  length  of  crank* 


27.  Fig.  23  shows  a  compound  mechanism.  The  link  d  is  the 
supporting  frame,  and  a  rotates  about  ad  in  the  direction  indicated, 
communicating  motion  to  c  through  the  slider  6,  so  that  c  vibrates 
about  cd.  The  link  e,  connected  to  c  by  a  turning  pair  at  ce,  causes 
f  to  slide  horizontally  on  another  part  of  the  frame  or  fixed  link  d. 
The  centre  of  the  crank-pin,  ab,  is  given  a  constant  linear  velocity, 
and  the  slider,/,  has  motion  toward  the  left  with  a  certain  mean 
velocity,  and  returns  toward  the  right  with  a  greater  mean  velocity. 
This  is  true  because  the  slider  /  moves  toward  the  left,  while  a 
moves  through  the  angle  «  ;  and  toward  the  right  while  a  moves 
through  the  angle  i^.  But  the  motion  of  a  is  uniform,  and  hence 
the  angular  movement  «  represents  more  time  than  the  angular 
movement /5 ;  and/,  therefore,  has  more  time  to  move  toward  the 
left,  than  it  has  to  move  through  the  same  space  toward  the  right. 
It  therefore  has  a  "quick  return"  motion. 

The  machine  is  driven  so  that  the  crank-pin  centre  moves  uni- 
formly, and  the  velocity,  at  all  points  of  its  stroke,  of  the  slider 
carrying  a  cutting  tool,  is  required.  The  problem,  therefore,  is  to  find 
the  relation  of  linear  velocities  of  ef  and  ah,  for  a  series  of  positions 
during  the  cycle  ;  and  to  draw^  the  diagram  of  velocity  of  ef. 

Solution.  —  ab  has  a  constant  known  linear  velocity.  The  point 
in  the  link  c  which  coincides,  for  the  instant,  with  ab,  receives 
motion  from  ah  ;  but  the  direction  of  its  motion  is  different  from 


24  MACHINE    DESIGN. 

that  of  ah,  because  ah  rotates  about  ad  while  the  coinciding  point  of 
c  rotates  about  cd.  If  ah- A  be  laid  off  representing  the  linear 
velocity  of  ah,  then  ah-B  will  represent  the  linear  velocity  of 
the  coinciding  point  of  the  link  r.  Let  the  latter  point  be 
called  X. 

Locate  cf,  at  the  intersection  of  e  with  the  line  cd  —  ad.  Now 
cf  and  X  are  both  points  in  the  link  c,  and  hence  their  linear  veloci- 
ties, relatively  to  the  fixed  link,  are  proportional  to  their  distances 
from  cd.  These  two  distances  may  be  measured  directly,  and  with 
the  known  value  of  linear  velocity  oi  x=^  ah-B,  give  three  known 
values  of  a  simple  proportion,  from  which  the  fourth  term,  the 
linear  velocity  of  rf,  may  be  found. 

Or,  if  the  line  BD  be  drawn  parallel  to  cd-ad,  the  triangle  B-D-ah 
is  similar  to  the  triangle  cd-cf-ah,  and  from  the  similarity  of  these 
triangles,  it  follows  that  BD  represents  the  linear  velocity  of  cf  on 
the  same  scale  that  ah-B  represents  the  linear  velocity  of  x.  Hence 
the  linear  velocity  of  cf,  for  the  assumed  position  of  the  mechanism, 
becomes  known.  But  since  cf  is  a  point  of  the  slider,  all  of  whose 
points  have  the  same  linear  velocity,  it  follows  that  the  linear 
velocity  of  cf  is  the  required  linear  velocity  of  the  slider. 

This  solution  may  be  made  for  as  many  positions  of  the  mech- 
anism as  are  necessary  to  locate  accurately  the  velocity  curve. 

Having  drawn  the  velocity  diagram,  suppose  that  it  is  required 

to  make  the  maximum  linear  velocity  of  the  slider  on  the  slow 

stroke  =  ^  feet  per  minute.     Then  the  linear  velocity  of  the  crank- 

,  .  max.  F/ of  slider        .,  .  ^         t 

pin  centre  ah  =zy  =  A  — — ^yz — - — .      If  r  —  the  crank  radius, 

VI  of  ah 


the  number  of  revolutions  per  minute 


JL 

When  this  mechanism  is  embodied  in  a  machine,  a  becomes  a 
crank  attached  to  a  shaft  whose  axis  is  at  ad.  The  shaft  turns  in 
bearings  provided  in  the  machine  frame.  The  crank  carries  a  pin 
whose  axis  is  at  ah,  and  this  pin  turns  in  a  bearing  in  the  sliding 
block  h.  The  link  c  becomes  a  lever  keyed  to  a  shaft  whose  axis  is 
at  cd.     This  lever  has  a  long  slot  in  which  the  block  h  slides.     The 


MOTION    IN    MECHANISMS.  25 

link  e  becomes  a  connecting-rod,  connected  both  to  c  and  /  by  pin 
and  bearing.  The  link/  becomes  the  "cutter  bar''  or  "ram"  of  a 
shaper :  the  part  which  carries  the  cutting  tool.  The  link  d 
becomes  the  frame  of  the  machine,  which  not  only  affords  support 
to  the  shafts  at  nd  and  cd,  and  the  guiding  su r f aces t for/,  but  also 
is  so  designed  as  to  afford  means  for  holding  the  pieces  to  be  planed, 
and  supports  the  feeding  mechanism. 

28.  Fig.  24  shows  another  compound  linkage,  d  is  fixed,  and  c 
rotates  uniformly  about  cd,  communicating  rotary  motion  to  a 
through  the  slider  h.  a  is  extended  past  ad  (the  part  extended 
being  in  another  parallel  plane),  and  moves  a  slider/ through  a 
link  e.  This  is  called  the  "  Whitworth  quick  return  mechanism." 
The  point  he  at  which  c  communicates  motion  to  a  moves  along  a, 
and  hence  the  radius  of  the  point  at  which  a  receives  a  constant 
linear  velocity  varies,  and  the  angular  velocity  of  a  must  vary 
inversely.  Hence  the  angular  velocity  of  a  is  a  maximum  when 
the  radius  is  a  minimum,  i.  e.,  when  a  and  c  are  vertical  down- 
ward ;  and  the  angular  velocity  of  a  is  minimum  when  the  radius 
is  a  maximum,  i.  e.,  when  a  and  c  are  vertical  upward. 

29.  Problem.  —  To  design  a  Whitworth  Quick  Return  for  a  given 

mean  VI  of  f  forward 
ratio. 


mean  VI  of/  returning' 


When  the  centre  of  the  crank-pin,  C,  reaches  A,  the  point  D  will 
coincide  with  B,  the  link  C  will  occupy  the  angular  position  cd-B, 
and  the  slider/  will  be  at  its  extreme  position  toward  the  left. 

When  the  point  C  reaches  F,  the  point  D  will  coincide  with  E, 
the  link  e  will  occupy  the  angular  position  cd-E,  and  the  slider  / 
will  be  at  its  extreme  position  toward  the  right. 

Obviously,  while  the  link  c  moves  over  from  the  position  cd-E  to 
the  position  cd-B,  the  slider  /  will  complete  its  forward  stroke. 
While  c  moves  under  from  ed-B  to  cd-E,  f  will  complete  the  return 
stroke.  The  link  c  moves  with  a  uniform  angular  velocity,  and 
hence  the  mean  velocity  of  /  forward  is  inversely  proportional  to 
the  angle  /^,  and  the  mean  velocity  of/  returning  is  inversely  pro- 


26  MACHINE    DESIGN. 

,  ,  ^  mean  VI  of  f  forward       « 

portional  to  «.     Or  zj-. — ^ — -. — =-. 

mean  Vloij  returning      ji 

For  the  design,  the  distance  cd-ad  must  be  known.     This  may 

usually  be  decided  on  from  the  limiting  sizes  of  the  journals  at  cd 

a      5 
and  ad.     Suppose  that  the  above  ratio  ==-=7^,   that   cd-ad ^Z'\ 

and  that  the  maximum   length  of   stroke  of  /=  12".  Locate  cd 

and  measure  off  vertically  downward  a  distance  equal  to  3",  thus 

locating  ad.     Draw  a  horizontal  line  through  ad.     The  point  ef  of 
the  slider/  will  move  along  this  line. 

a       5 
Since  7;=^  ,  and  «  + /5  =  360'', 

.-.  a-=150°andr^  =  210°. 
Lay  off  «  from  cd  as  a  centre,  so  that  the  vertical  line  through  cd 
bisects  it.     Draw  a  circle  through  B  with  cd  as  a  centre,  B  being 
the  point  of  intersection  of  the  bounding  line  of  «  with  a  horizontal 
through  ad.     The  length  of  the  link  c^cd-B. 

The  radius  ad-C  must  equal  the  travel  of /-^2  =  6".  This 
radius  is  made  adjustable,  so  that  the  length  of  stroke  may  be 
varied.  The  connecting-rod,  e,  may  be  made  of  any  convenient 
length. 

30.  Problem.  —  To  draw  the  velocity  diagram  of  the  slider  /  of 
the  Whitworth  Quick  Return.  The  point  he,  Fig.  25,  has  a  known 
constant  linear  velocity,  and  its  direction  of  motion  is  always  at 
right  angles  to  a  line  joining  it  to  cd.  The  point  of  the  link  a, 
which  coincides  in  this  position  of  the  mechanism  with  he,  receives 
motion  from  he,  but  its  direction  of  motion  is  at  right  angles  to  the 
line  hc-ad.  If  bc-A  represent  the  linear  velocity  of  he,  its  projection 
upon  hc-ad  extended  will  represent-  the  linear  velocity  of  the  point 
of  a  which  coincides  with  he.  Call  this  point  x.  The  centro  af 
may  be  considered  as  a  point  in  a,  and  its  linear  velocity  relatively 
to  d,  when  so  considered,  is  proportional  to  its  distance  from  ad. 

VI  of  af      ad-af 
Hence  y.,    ,     =     ■,  ,     . 

V I  of  X       ad-bc 


/^^   Of  THK 

■0H  ITER  SIT  7; 


MOTION    IN    MECHANISMS.  27 

But  the  triangles  ad-cd-bc,  and  B-C-bc  are  similar.     Hence 

VI  of  af_  BC 
VI  of  X  ~~  B-bc  ' 

This  means  that  BC  represents  the  linear  velocity  of  af  upon  the 
same  scale  that  B-bc  represents  the  linear  velocity  of  x.  But  af  is  a 
point  in  /,  and  all  points  in  /  have  the  same  linear  velocity  ;  hence 
BC  represents  the  linear  velocity  of  the  slider/,  for  the  given  posi- 
tion of  the  mechanism,  and  it  may  be  laid  off  as  an  ordinate  of  the 
velocity  curve.  This  solution  may  be  made  for  as  many  positions 
as  are  required  to  locate  accurately  the  entire  velocity  curve  for  a 
cycle  of  the  mechanism. 


CHAPTER   III. 


ENERGY    IN    MACHINES. 


31.  The  subject  of  motion  and  velocity,  in  certain  simple 
machines,  has  been  treated  and  illustrated.  It  remains  now  to 
consider  the  passage  of  energy  through  similar  machines.  From 
this  the  solution  of  force  problems  will  follow. 

During  the  passage  of  energy  through  a  machine,  or  chain  of 
machines,  any  one.  or  all,  of  four  changes  may  occur. 

I.  The  energy  may  be  transferred  in  space.  Example.  —  Energy 
is  received  at  one  end  of  a  shaft  and  transferred  to  the  other  end, 
where  it  is  received  and  utilized  by  a  machine. 

II.  The  energy  may  be  converted  into  another  form.  Ey ampler. 
—  (a)  Heat  energy  into  mechanical  energy  by  the  steam  engine 
machine  chain.  (6)  Mechanical  energy  into  heat  by  friction. 
{c)  Mechanical  energy  into  electrical  energy,  as  in  a  dynamo- 
electric  machine  ;  or  electrical  energy  into  mechanical  energy  in 
the  electric  motor,  etc. 

III.  Energy  is  the  product  of  a  force  factor  and  a  space  factor. 
Energy  per  unit  time,  or  rate  of  doing  work,  is  the  product  of  a 
force  factor  and  a  velocity  factor  ;  since  velocity  is  space  per  unit 
time.  Either  factor  may  be  changed  at  the  expense  of  the  other  ; 
i.  e.,  velocity  may  be  changed,  if  accompanied  by  such  a  change  of 
force  that  the  energy  per  unit  time  remains  constant.  Correspond- 
ingly, force  may  be  changed  at  the  expense  of  velocity,  energy  per 
unit  time  being  constant.  Example.  —  A  belt  transmits  6000  foot- 
pounds of  energy  per  minute  to  a  machine.     The  belt  velocity  is 


ENERGY    IN    MACHINES.  29 

120  feet  per  minute,  and  the  force  exerted  is  50  lbs.  Frictional 
resistance  is  neglected.  A  cutting  tool  in  the  machine  does  useful 
work  ;  its  velocity  is  20  feet  per  minute,  and  the  resistance  to  cut- 
ting is  800  lbs.  Then,  energy  received  per  minute  =  120  X  50  = 
6000  foot-pounds  ;  and  energy  delivered  per  minute  =  20  X  800  = 
(iOOO  foot-pounds.  The  energ}^  received  therefore  equals  the  energy 
delivered.  But  the  velocity  and  force  factors  are  quite  different  in 
the  two  cases. 

IV.  Energy  may  be  transferred  in  time.  In  many  machines 
the  energy  received  at  every  instant  equals  that  delivered.  There 
are  many  cases,  however,  where  there  is  a  periodical  demand  for 
work,  i.  e.,  a  fluctuation  in  the  rate  of  doing  work  ;  while  energy 
can  only  be  supplied  at  the  average  rate.  Or  there  may  be  a  uni- 
form rate  of  doing  work,  and  a  fluctuating  rate  of  supplying  energy. 
In  such  cases  means  are  provided  in  the  machine,  or  chain  of 
machines,  for  the  storing  of  energy  till  it  is  needed.  In  other  words, 
energy  is  transferred  in  time.  Examples.  —  (a)  In  the  steam  engine 
there  is  a  varying  rate  of  supplying  energy  during  each  stroke, 
while  there  is  (in  general)  a  uniform  rate  of  doing  work.  There  is, 
therefore,  a  periodical  excess  and  deficiency  of  effort.  A  heavy 
wheel  on  the  main  shaft  absorbs  the  excess  of  energy  with  increased 
velocity,  and  gives  it  out  again  with  reduced  velocity,  when  the 
effort  is  deficient,  (h)  A  pump  delivers  water  into  a  pipe  system 
under  pressure.  The  water  is  used  in  a  hydraulic  press,  whose 
action  is  periodic,  and  beyond  the  capacity  of  the  pump.  A 
hydraulic  accumulator  is  attached  to  the  pipe  system,  and  while 
the  press  is  idle  the  pump  slowly  raises  the  accumulator  weight, 
thereby  storing  potential  energy,  which  is  given  out  rapidly  b)^  the 
descending  weight  for  a  short  time  while  the  press  acts,  (r)  A 
dynamo-electric  machine  is  run  by  a  steam  engine,  and  the  electri- 
cal energy  is  delivered  and  stored  in  storage  batteries,  upon  which 
there  is  a  periodical  demand.  In  this  case,  as  well  as  in  case  (?>), 
there  is  a  transformation  of  energy  as  well  as  a  transfer  in  time. 

32.  Force  Problems.  —  Suppose  the  slider  crank  mechanism  in 
Fig.  26   to  represent  a  shaping  machine  ;  the  velocity  diagram  of 


30  MACHINE    DESIGN. 

the  slider  being  drawn.  The  resistance  offered  to  cutting  metal 
during  the  forward  stroke  must  be  overcome.  This  resistance  may 
be  assumed  constant.  Throughout  the  cutting  stroke  there  is  a 
constantly  varying  rate  of  doing  work.  This  is  because  the  rate  of 
doing  work  =  resisting  force  (constant) X  velocity  (varying).  This 
product  is  constantly  varying,  and  is  a  maximum  when  the  slider's 
velocity  is  a  maximum.  The  slider  must  be  driven  by  means  of 
energy  transmitted  through  the  crank  a.  The  maximum  rate  at 
which  energy  must  be  supplied,  equals  the  maximum  rate  of  doing 
work  at  the  slider.  Draw  the  mechanism  in  the  position  of  max- 
imum velocity  of  slider  ;  i.  e.,  locate  the  centre  of  the  slider-pin  at 
the  base  of  the  maximum  ordinate  of  the  velocity  diagram,  and 
draw  h  and  a  in  their  corresponding  positions.  The  slider's  known 
velocity  is  represented  by  y,  and  the  crank-pin's  required  velocity 
is  represented  by  z  on  the  same  scale.  Hence  the  value  of  x  becomes 
known  by  simple  proportion.  The  rate  of  doing  work  must  be  the 
same  at  c  and  at  ah  ( neglecting  friction )  .*  Hence  Rv^  =  Fv.^,  in  which 
R  and  i\  represent  the  force  and  velocity  factors  at  c  ;  and  F  and  v.^ 
represent  the  force  and  velocity  factors  at  ah.  R  and  v^  are  known 
from    the   conditions  of  the  problem,   and  v.^^  is  found    as   above. 

Rv 
Hence,  i^may  be  found,  ^ — ^  =  force  which,  applied  tangentially 

to  the  crank-pin  centre,  will  overcome  the  maximum  resistance  of 
the  machine.  In  all  other  positions  of  the  cutting  stroke  the  rate 
of  doing  work  is  less,  and  F  would  be  less.  But  it  is  necessary  to 
provide  driving  mechanism  capable  of  overcoming  the  maximum 
resistance,  when  no  fly-wheel  is  used.  If  now  F  be  multiplied  by 
the  crank  radius,  the  product  equals  the  maximum  torsional 
moment  (=M)  required  to  drive  the  machine.  If  the  energy  is 
received  on  some  different  radius,  as  in  case  of  gear  or  belt  trans- 
mission, the  maximum  driving  force  =  3/  -^  the  new  radius.  During 
the  return  stroke  the  cutting  tool  is  idle,  and  it  is  only  necessary  to 

*The  effect  of  acceleration  to  redistribute  energy  is  zero  in  this  position, 
because  the  acceleration  of  the  slider  at  maximum  velocity  is  zero,  and  the 
angular  acceleration  of  h  can  only  produce  pressure  in  the  journal  at  ad. 


ENEHGY    IN    MACHINES.  31 

overcome  the  frictional  resistance  to  motion  of  the  bearing  surfaces. 
Hence,  the  return  stroke  is  not  considered  in  designing  the  driving 
mechanism.  When  the  method  of  driving  this  machine  is  decided 
on,  the  capacity  of  the  driving  mechanism  must  be  such  that  it 
shall  be  capable  of  supplying  to  the  crank  shaft  the  torsional 
driving  moment  M,  determined  as  above. 

This  method  applies  as  well  to  the  quick  return  mechanisms 
given.  In  each,  when  the  velocity  diagram  is  drawn,  the  vector  of 
the  maximum  linear  velocity  of  the  slider,  =  Zi,  and  of  the  constant 
linear  velocity  of  the  crank-pin  centre,  =  Z2,  are  known,  and  the 
velocities  corresponding,  V^  and  V^,  are  also  known,  from  the  scale 
of  velocities.  The  rate  of  doing  work  at  the  slider  and  at  the  crank- 
pin  centre  is  the  same,  friction  being  neglected.  Hence  Ri\  =  Fv.^, 
or,  since  the  vector  lengths  are  proportional  to  the  velocities  they 

represent,  RL^  ==  FL.^ ;  and  F==  -j-^ .     Therefore  the  resistance  to  the 

slider's  motion  =R,  on  the  cutting  stroke,  multiplied  by  the  ratio  of 

linear  velocity  vectors,  ~,  of   slider  and  crank-pin,  equals  F,  the 

maximum  force  that  must  be  applied  tangentially  at  the  crank- 
pin  centre  to  insure  motion.  F  multiplied  by  the  crank  radius 
=  maximum  torsional  driving  moment  required  by  the  crank 
shaft.  If  R  is  varying,  and  known,  find  where  Rv,  the  rate  of 
doing  work,  is  a  maximum,  and  solve  for  that  position  in  the 
same  way  as  above. 

33.  Force  Problems  —  Continued.  —  In  the  usual  type  of  steam 
engine  the  slider  crank  mechanism  is  used,  but  energy  is  supplied 
to  the  slider  (which  represents  piston,  piston-rod,  and  cross-head), 
and  the  resistance  opposes  the  rotation  of  the  crank  and  attached 
shaft.  In  any  position  of  the  mechanism.  Fig.  28,  force  applied  to 
the  crank-pin  through  the  connecting-rod,  may  be  resolved  into 
two  components,  one  radial  and  one  tangential.  The  tangential 
component  tends  to  produce  rotation  ;  the  radial  component  pro- 
duces pressure  between  the  surfaces  of  the  shaft  journal  and  its 
bearing.    The  tangential  component  is  a  maximum  when  the  angle 


32  MACHINE    DESIGN. 

between  crank  and  connecting-rod  equals  90°;  and  it  becomes  zero 
when  C  reaches  A  or  B.  If  there  is  a  uniform  resistance,  the  rate  of 
doing  work  is  constant.  Hence,  since  the  energy  is  supplied  at  a 
varying  rate,  it  follows  that  during  part  of  the  revolution  the  effort 
is  greater  than  the  resistance  ;  while  during  the  remaining  portion 
of  the  revolution  the  effort  is  less  than  the  resistance.  When  effort 
is  less  than  resistance,  the  machine  will  stop  unless  other  means 
are  provided  to  maintain  motion.  A  "fly-wheel"  is  keyed  to  the 
shaft,  and  this  wheel,  because  of  slight  variations  of  velocity,  alter- 
nately stores  and  gives  out  the  excess  and  deficiency  of  energy  of 
the  effort,  thereby  adapting  it  to  the  constant  work  to  be  done. 

34.  Problem.  —  Given  length  of  stroke  of  the  slider  of  a  steam 
engine  slider  crank  mechanism,  the  required  horse-power,  or  rate  of 
doing  work,  and  number  of  revolutions.  Required  the  total  mean 
pressure  that  must  be  applied  to  the  piston. 

Let  L  =  length  of  stroke  =  1  foot ; 

HP  =  horse-power  =  20  ; 
N  =  revolutions  per  minute  =  200  ; 
F  =  required  mean  force  on  piston. 

Then  NXL^=  200  feet  per  minute ^=  mean  velocity  of  slider  =V. 

Now,  the  mean  rate  of  doing  work  in  the  cylinder  and  at  the  main 

shaft  during  each  stroke  is  the  same  (friction  neglected);  hence  FV 

-=  HP  X  33000, 

^      HP  X  33000      20X33000      ^^^  ,. 
F- ^ -= 1^ -=3300  lbs. 

35.  In  the  slider  crank  chain,  the  velocity  of  the  slider  neces- 
sarily varies  from  zero,  at  the  ends  of  its  stroke,  to  a  maximum 
value  near  mid-stroke.  The  mass  of  the  slider  and  attached  parts 
is  therefore  positively  and  negatively  accelerated  each  stroke. 
When  a  mass  is  positively  accelerated  it  stores  energy  ;  and  when 
it  is  negatively  accelerated  it  gives  out  energy.  The  amount  of  this 
energy,  stored  or  given  out,  depends  upon  the  mass  and  the  accel- 
eration. The  slider  stores  energy  during  the  first  part  of  its  stroke, 
and  gives  it  out  during  the  second  part  of  its  stroke.     While,  there- 


ENKKGY    IN    MACHINES.  88 

fore,  it  gives  out  all  the  energy  it  receives,  it  gives  it  out  differently 
distributed.  In  order  to  find  exactly  how  the  energy  is  distributed, 
it  is  necessary  to  find  the  acceleration  throughout  the  slider's 
stroke.  This  may  be  done  as  follows  :  Fig.  27  A  shows  the  velo- 
city  diagram   of  the  slider  of  a  slider  crank   mechanism,  for  the 

forward  stroke.     The  acceleration  required  at  any  point       -r—,    in 

which  Jr  is  the  increase  in  velocity  during  any  interval  of  time 
J^,  assuming  that  the  increase  in  velocity  becomes  constant  at  that 
point.  Lay  off  the  horizontal  line  OP~MN.  Divide  OP  into  as 
many  equal  parts  as  there  are  unequal  parts  in  MN.  These  divis- 
ions may  each  represent  Jf.  At  m  erect  the  ordinate  mn  =  m^n^^, 
and  at  o  erect  the  ordinate  op  OiPi.  Continue  this  construction 
throughout  OP,  and  draw  a  curve  through  the  upper  extremities  of 
the  ordinates.  H  is  a  velocity  diagram  on  a  "  time  base."  At  O 
draw  the  tangent  OT  to  the  curve.  If  the  increase  in  velocity  were 
constant  during  the  time  interval  represented  by  Om,  the 
increment  of  velocity  would  be  represented  by  mT.  Therefore 
rM-7Ms  proportional  to  the  acceleration  at  the  point  0,  and  may  be 
laid  off  as  an  ordinate  of  an  acceleration  diagram  C  Thus  Qa  -= 
inT.  The  divisions  of  QR  are  the  same  as  those  of  MN  \  i.  e.,  they 
represent  positinna  of  the  slider.  This  construction  may  be  repeateil 
for  the  other  divisions  of  the  curve  B.  Thus  at  n  the  tangent  n7\ 
and  horizontal  nq  are  drawn,  and  qT^  is  proportional  to  the  accel- 
eration at  n,  and  is  laid  off  as  an  ordinate  he  of  the  acceleration 
diagram.  To  find  the  value  in  acceleration  units  of  Qa,  mT  is  read 
off  in  velocity  units  ^^Jr,  by  the  scale  of  ordinates  of  the  velocity 
diagram.'   This  value  is  divided  by  J^  the  time  increment  corres- 

Jv 
ponding  to  Om.     The  result  of  this  division    ^    -  acceleration  at 

M  in  acceleration  units.  Jt  ^=  the  time  of  one  stroke,  or  of  one- 
half  revolution  of  the  crank,  divided  by  the  number  of  divisions  in 
OP.     If  the  linear  velocity  of  the  centre  of  the  crank-pin,  =  V,  be 

MN 
represented  by  the  length  of  the  crank  radius       -^r— =  r,  then   the 

5  ^ 


34  MACHINE    DESIGN. 

scale  of  velocities,  or  velocity  in  feet  per  second  for  1"  of  ordinate 

V     izDN 

=^  —  =  ^TTp:.     D  is  the  actual  diameter  of  the  crank  circle,  and  r  is 
r        r  hi) 

the  crank  radius  measured  on  the  figure.  If  the  weight,  W,  of  parts 
accelerated  is  known,  the  force,  F,  necessary  to  produce  the  accel- 
eration at  any  slider  position  may  be  found  from  the  fundamental 
formula  of  mechanics 

9 
p  being  the  acceleration  corresponding  to  the  position  considered. 
If  the  ordinates  of  the  acceleration  diagram  be  taken  as  represent- 
ing the  forces  which  produce  the  acceleration,  the  diagram  will  have 
force  ordinates  and  space  abscissae,  and  areas  will  represent  work. 
Thus,  Qas  represents  the  work  stored  during  acceleration,  and  Rsd 
represents  the  work  given  out  during  retardation.  Let  MN,  Fig.  29, 
represent  the  length  of  the  slider's  stroke,  and  NC  the  resistance  of 
cutting  (uniform);  then  energy  to  do  cutting  per  stroke  is  repres- 
ented by  the  area  MBCN.  But  during  the  early  part  of  the  stroke 
the  reciprocating  parts  must  be  accelerated,  and  the  force  necessary 
at  the  beginning,  found  as  above,  =5Z>.  The  driving  gear  must, 
therefore,  be  able  to  overcome  resistance  equal  to  MB  -\-  BD.  The 
acceleration,  and  hence  the  accelerating  force,  decreases  as  the 
slider  advances,  becoming  zero  at  E,  From  E  on  the  acceleration 
becomes  negative,  and  hence  the  slider  gives  out  energy  and  helps 
to  overcome  the  resistance,  and  the  driving  gear  has  only  to  furnish 
energy  represented  by  the  area  AEFN,  though  the  work  really 
done  against  resistance  equals  that  represented  by  the  area  CEFN. 
The  energy  represented  by  the  difference  of  these  areas,  =  ^CjE',  is 
that  which  is  stored  in  the  slider's  mass  during  acceleration.  Since 
by  the  law  of  conservation  of  energy,  energy  given  out  per  cycle  =: 
that  received,  it  follows  that  area  ^C£'  =  area  DEB,  and  area 
BCMN=ADMN.  This  redistribution  of  energy  would  seem  to 
modify  the  problem  on  page  30,  since  that  problem  is  based  on  the 
assumption  of  uniform  resistance  during  cutting  stroke.  The 
position   of   maximum  velocity  of   slider,   however,  corresponds  to 


enp:rgy  in  machines.  35 

acceleration  ^=0.  The  maximum  rate  of  doing  work,  and  the  cor- 
responding torsional  driving  moment  at  the  crank  shaft  would 
probably  correspond  to  the  same  position,  and  would  not  be  materi- 
ally changed.  In  such  machines  as  shapers,  the  acceleration  and 
weight  of  slider  are  so  small  that  the  redistribution  of  energy  is 
unimportant. 

36.  Solution  of  the  force  problem  in  the  steam  engine  slider  crank 
mechanism  ;  slider  represents  piston  with  its  rod,  and  the  cross- 
head. —  The  steam  acts  upon  the  piston  with  a  pressure  which 
varies  during  the  stroke.  The  pressure  is  redistributed  before 
reaching  the  cross-head  pin,  because  the  reciprocating  parts  are 
accelerated  in  the  first  part  of  the  stroke,  with  accompanying 
storing  of  energy  and  reduction  of  pressure  on  the  cross-head 
pin  ;  and  retarded  in  the  second  part  of  the  stroke,  with  accom- 
panying giving  out  of  energy  and  increase  of  pressure  on  the 
cross-head  pin.  Let  the  ordinates  of  the  full  line  diagram  above 
OX,  Fig.  30  A,  represent  the  effective  pressure  on  the  piston 
throughout  a  stroke.  B  is  the  velocity  diagram  of  slider.  Find 
the  acceleration  throughout  stroke,  and  from  this  and  the  known 
value  of  weight  of  slider,  find  the  force  due  to  acceleration.  Draw 
diagram  C,  whose  ordinates  represent  the  force  due  to  accelera- 
tion, upon  the  same  force  scale  used  in  A,  Lay  off  this  diagram 
on  OX  as  a  base  line,  thereby  locating  the  dotted  line.  Thu 
vertical  ordinates  between  this  dotted  line  and  the  upper  line  of 
A  represent  the  pressure  applied  to  the  cross-head  pin.  These 
ordinates  may  be  laid  off  from  a  horizontal  base  line,  giving 
D.  The  product  of  the  values  of  the  corresponding  ordinates 
of  B  and  D  =  the  rate  of  doing  work  throughout  the  stroke.  Thus 
the  value  of  GH  in  pounds  X  value  of  EF  in  feet  per  second 
— -  the  rate  of  doing  work  in  foot-pounds  per  second  upon  the 
cross-head  pin,  when  the  centre  of  the  cross-head  pin  is  at  E. 
The  rate  of  doing  work  at  the  crank-pin  is  the  same  as  at  the  cross- 
head  pin.  Hence  dividing  this  rate  of  doing  work,  =  EF  X  GH, 
by  the  constant  tangential  velocity  of  the  crank-pin  centre,  gives 
the  force  acting  tangentially  on   the  crank-pin   to   produce  rota- 


86  MACHINK    DESIGN. 

tion.  The  tangential  forces  acting  throughout  a  half  revolution 
of  the  crank  may  be  thus  found,  and  plotted  upon  a  horizontal 
base  line=:  length  of  half  the  crank  circle,  Fig.  81.  The  work  done 
upon  the  piston,  cross-head  pin,  and  crank  during  a  piston  stroke  is 
the  same.  Hence  the  areas  of  A  and  /),  Fig.  80,  are  equal  to 
each  other,  and  to  the  area  of  the  diagram.  Fig.  81.  The  forces 
acting  along  the  connecting-rod  for  all  positions  during  the  piston 
stroke,  may  be  found  by  drawing  force  triangles  with  one  side  hor- 
izontal, one  vertical,  and  one  parallel  to  position  of  connecting-rod 
axis,  the  horizontal  side  being  equal  to  the  corresponding  ordinate 
of  D.  The  vertical  sides  of  these  triangles  will  represent  the  guide 
reaction,  while  the  side  parallel  to  the  connecting-rod  axis  repre- 
sents the  force  transmitted  by  the  connecting-rod. 


5^. 


or  TOM 


e 


Z 


lu 


D. 


S2?L 


ce 


^c     c Li 


^ 


r/g.^32 


/ 


UHIVBR 


chaptp:r  IV. 

PARALLEL    OR    STRAIGHT   LINE    MOTIONS. 

37.  Rectilinear  motion  in  machines  is  usually  obtained  by 
means  of  prismatic  guides.  It  is  sometimes  necessary,  however,  to 
accomplish  the  same  result  by. linkages. 

A  general  n)ethod  of  design,  which  is  applicable  in  many  cases, 
will  be  given.  In  Fig.  82,  d  is  the  fixed  link,  and  a  is  connected 
with  it  by  a  sliding  pair,  a,  b,  c,  and  e  are  connected  by  turning 
pairs,  as  shown.  The  constrainment  is  not  complete  because  B  is 
free  to  move  in  any  direction,  and  its  motion  would,  therefore, 
depend  upon  the  force  producing  it.  It  is  required  that  the  point 
B  shall  move  in  a  straight  line  parallel  to  a.  Suppose  that  B  is 
caused  to  move  along  the  required  line  ;  then  any  point  of  the  link 
r,  as  A,  will  describe  some  curve,  FAE.  If  a  pin  be  attached  to  i\ 
with  its  axis  at  A,  and  a  curved  slot  fitting  the  pin,  with  its  sides 
parallel  to  FAE,  be  attached  to  d,  as  in  Fig.  83,  it  follows  that  B 
can  only  move  in  the  required  straight  line.  This  is  the  mechanism 
of  the  Tabor  Steam  Engine  Indicator. 

The  curve  described  by  A  might  approximate  a  circular  arc  whose 
centre  could  be  located,  say  at  0,  Fig.  88.  Then  the  curved  slot 
might  be  replaced  by  a  link,  attached  to  d  and  c  by  turning  pairs 
at  0  and  A.  This  gives  B  approximately  the  required  motion. 
This  is  the  mechanism  of  the  Thompson  Steam  Engine  Indicator. 

If,  while  the  point  B  is  caused  to  move  in  the  required  straight 
line,  a  point  in  6,  as  P,  Fig.  82,  were  chosen,  it  would  be  found  to 
describe  a  curve  which  would  approximate  a  circular  arc,  whose 
centre,  0,  and    radius,  =  r.  could    be    found.     Let    the  link  whose 


38  MACHINE    DESIGN. 

length  =  r  he  attached  to  d  and  h  by  turning  pairs  whose  axes  are 
at  0  and  P,  and  the  motion  of  B  will  be  approximately  the 
required  motion.  This  is  the  mechanism  of  the  Crosby  Steam 
Engine  Indicator. 

38.  Problem.  — In  Fig.  34,  B  is  the  fixed  axis  of  a  counter-shaft ; 
C  is  the  axis  of  another  shaft  which  is  free  to  rotate  about  B.  D  is 
the  axis  of  a  circular  saw  which  is  free  to  move  in  any  direction. 
It  is  required  to  constrain  D  to  move  in  the  straight  line  EF.  If  D 
be  moved  along  EF,  a  tracing  point  fixed  at  A  in  the  link  CD  will 
describe  an  approximate  circular  arc,  HAK,  whose  centre  may  be 
found  at.O.  A  link  whose  length  is  OA  may  be  connected  to  the 
fixed  link,  and  to  the  link  CD  by  means  of  turning  pairs  at  0  and 
A.  D  will  then  be  constrained  to  move  approximately  along  EF. 
A  curved  slot  and  pin  could  be  used,  and  the  motion  would  be 
exact.* 

*  Descriptions  of  many  varieties  of  parallel  motions  may  be  found  in  Ran- 
kine's  "  Machinery  and  Millwork  "  ;  Weisbach's  "  Mechanics  of  Engineering," 
Vol.  Ill,  ** Mechanics  of  the  Machinery  of  Transmission";  Kennedy's 
*'  Mechanics  of  Machinery." 


^^^^^ 


0*  THB 


CHAPTER   V. 

TOOTHED   WHEELS,    OR   GEARS. 

39.  When  toothed  wheels  are  used  to  communicate  motion,  the 
motion  elements  are  the  tooth  surfaces.  The  contact  of  these  sur- 
faces with  each  other  is  line  contact.  Such  pairs  of  motion  elements 
are  called  higher  pairs,  to  distinguish  them  from  lower  pairs,  which 
are  in  contact  throughout  their  entire  surface.  Fig.  35  shows  the 
simplest  toothed  wheel  mechanism.  There  are  three  links,  a,  h,  and 
c,  and  therefore  three  centros  ab,  be,  and  ac.  These  centres  must,  as 
heretofore  explained,  lie  in  the  same  straight  line,  ac  and  ab  are 
the  centres  of  the  turning  pairs  connecting  c  and  b  to  a.  It  is  re- 
quired to  locate  be  on  the  line  of  centres. 

When  the  gear  e  is  caused  to  rotate  uniformly  with  a  certain 
angular  velocity,  i.  e.,  at  the  rate  of  m  revolutions  per  minute,  it  is 
required  to  cause  the  gear  b  to  rotate  uniformly  at  a  rate  of  n  revo- 
lutions per  minute.     The  angular  velocity-ratio  is  therefore  con- 

stant,  and  =  — .     The  centro  be  is  a  point  on  the  line  of  centres 

which  has  the  same  linear  velocity  whether  it  is  considered  as  a 
point  in  b  or  e.  The  linear  velocity  of  this  point  be  in  b  =  2r^R{ti ; 
and  the  linear  velocity  of  the  same  point  in  c  =  lTiR.^m ;  in  which 
i?,  =  radius  of  be  in  ft,  and  i?.^  =  radius  of  be  in  e.  But  this  linear 
velocity  must  be  the  same  in  both  cases,  and  hence  the  above 
expressions  may  be  equated  thus  : 

,  R,      m 

whence  — i  =  —  . 

Ri      ^ 


40  MACHINE    DESIGN. 

Hence  he  is  located  by  dividing  the  line  of  centres  into  parts  which 

are  to  each  other  inversely  as  the  angular  velocities  of  the  gears. 

Thus,  let  ah  and  ac,  Fig.  86,  be  the  centres  of  a  pair  of  gears 

771 

whose  angular  velocity  ratio  =  -.      Draw  the  line  of  centres  ;  divide 

n 

into  m  i- n  equal  parts  ;   m  of  these  from  ab  toward   the  right,  or  ?» 

from  ar  toward  the  left,  will  locate  he.     Draw  circles  through  hr, 

with  aft* and  ar  as  centres.     These  circles  are  the  centroids  of  he  and 

are  called  pitch  cirdeH.     It   has  been    already  explained  that  any 

motion  may  be  reproduced  by  rolling  the  centroids  of  that  motion 

upon  each  other  without  slipping.     Therefore  the  motion  of  gears 

is  the  same  as  that  which  would  result  from  the  rolling  together  of 

the  pitch  circles  (or  cylinders)  without  slipping.     In    fact,   these 

pitch  cylinders  themselves   might  be,  and  sometimes  are,  used  ft)r 

transmitting  motion  of  rotation.     Slipping,  however,  is  apt  to  occur, 

and  hence  these  "  friction   gears "  cannot   be  used  if  no    variation 

from  the  given  velocity  ratio  is  allowable.     Hence,  teeth  are  formed 

on  the  wheels  which  engage  with  each  other,  to  prevent  slipping. 

40.  Teeth  of  almost  any  form  may  be  used,  and  the  average 
velocity  will  be  right.  But  if  the  forms  are  not  correct  there  will 
be  continual  variations  of  velocity  ratio  between  a  minimum  and 
maximum  value.  These  variations  are  in  many  cases  unallowable, 
and  in  all  cases  undesirable.  It  is  necessary  therefore  to  study 
tooth  outlines  which  shall  serve  for  the  transmission  of  a  constant 
velocity  ratio. 

The  centro  of  relative  motion  of  the  two  gears  must  remain  in  a 
constant  position  in  order  that  the  velocity  ratio  shall  be  constant. 
The  essential  condition  for  constant  velocity  ratio  i»,  therefore^  that  the 
position  of  the  centro  of  relative  motion  of  the  gears  shall  remain  un- 
changed. If  A  and  /?,  Fig.  88,  are  tooth  surfaces  in  contact  at  a, 
their  only  possible  relative  motion,  if  they  remain  in  Qontact,  is 
slipping  motion  along  the  tangent  CD.  The  centro  of  this  motion 
must  be  in  EF,  a  normal  to  the  tooth  surfaces  at  the  point  of  con- 
tact. If  these  be  supposed  to  be  teeth  of  a  pair  of  gears,  h  and  c, 
whose  required  velocity   ratio  is  known,  and  whose   centro,  be,  is 


^^-^ 


Ot  THS 


^HHIVBRSITY 


£1 


m."^ 


TOOTHED    WHEELS,    OR   GEARS.  41 

therefore  located,  then  in  order  that  the  motion  communicated  from 
one  gear  to  the  other  through  the  point  of  contact,  a,  shall  be  the 
required  motion,  it  is  necessary  that  the  centro  of  the  relative  motion 
of  the  teeth  shall  coincide  with  be. 

lUustration.  —  In  Fig.  38,  let  ac  and  ah  be  centres  of  rotation 
of  bodies  h  and  c,  and  the  required  velocity  ratio  is  such  that  the 
centro  of  b  and  c  falls  at  be.  Contact  between  b  and  c  is  at  p.  The 
only  possible  relative  motion  if  these  surfaces  remain  in  contact  is 
slipping  along  CD  ;  hence  the  centro  of  this  motion  must  be  on  EF, 
the  normal  to  the  tooth  surfaces  at  the  point  of  contact.  But  it  must 
also  be  on  the  same  straight  line  with  ac  and  ab  ;  hence  it  is  at  be, 
and  the  motion  transmitted  for  the  instant,  at  the  point  p,  is  the 
required  motion,  because  its  centro  is  at  be.  But  the  curves  touch- 
ing at  p,  might  be  of  such  form  that  their  common  normal  at  p 
would  intersect  the  line  of  centres  at  some  other  point,  as  K,  which 
would  then  become  the  centro  of  the  motion  of  b  and  c  for  the 
instant,  and  would  correspond  to  the  transmission  of  a  different 
motion.  The  essential  condition  to  be  fulfilled  by  tooth  outlines,  in 
order  that  a  constant  velocity  ratio  may  be  maintained,  may  there- 
fore be  stated  as  follows  :  The  tooth  outlines  must  be  sueh  that  their 
normal  at  the  point  of  eontact  shall  always  pass  through  the  eentro 
corresponding  to  the  required  velocity  ratio. 

41.  Having  given  any  curve  that  will  serve  for  a  tooth  outline 
in  one  gear,  the  corresponding  curve  may  be  found  in  the  other 
gear,  which  will  engage  with  the  given  curve  and  transmit  a  con- 

stant  velocity  ratio.     Let  —  be  the  given  velocity  ratio.     Draw  the 

n 

line  of  centres  AB,  Pig.  89.  Let  P  be  the  "  pitch  point,"  i.  e.,  the 
point  of  contact  of  the  pitch  circles  or  the  centro  of  relative  motion 
of  the  two  gears.  To  the  right  from  P  lay  off  a  distance  PB  =  m  ; 
from  P  toward  the  left  lay  off  PA=n.  A  and  B  will  then  be  the 
required  centres  of  the  wheels,  and  the  pitch  circles  may  be  drawn 
through  P.  Let  abc  be  any  given  curve  on  the  wheel  A.  It  is 
required  to  find  the  curve  in  B  which  shall  engage  with  abc  to 
transmit  the  constant  velocity  ratio  required.     A  normal  to  the 

6 


42  MACHINE    DESIGN. 

point  of  contact  must  pass  through  the  centro.  If,  therefore,  any 
point,  as  a,  be  taken  in  the  given  curve,  and  a  normal  to  the  curve 
at  that  point  be  drawn,  as  a«,  then  when  a  is  the  point  of  contact, 
«  will  coincide  with  P.  Also,  if  cy  is  a  normal  to  the  curve  at  c, 
then  Y  will  coincide  with  P  when  c  is  the  point  of  contact  between 
the  gears ;  and  since  h  is  in  the  pitch  line,  it  will  itself  coincide 
with  P  when  it  is  the  point  of  contact. '  Suppose  now  that  A  and  B 
are  discs  of  cardboard,  that  A  overlaps  B,  and  that  a  thread  is 
stretched  to  indicate  the  centre  line  AB.  Suppose  also  that  they 
can  be  rotated  so  that  the  pitch  circles  roll  on  each  other  without 
slipping.  Roll  the  discs  till  «  reaches  P,  and  prick  a  through  upon 
B  ;  then  make  h  coincide  with  P,  and  prick  it  through  ;  then  make 
y  coincide  with  P,  and  prick  c  through.  This  will  give  three  points 
in  the  required  curve  in  P,  and  through  these  the  curve  may  be 
drawn.  The  curve  could,  of  course,  be  more  accurately  located  by 
using  more  points.  The  points  of  the  curve  in  B  might  be  located 
geometrically. 

Many  curves  could  be  drawn  that  would  not  serve  for  tooth  out- 
lines ;  but,  given  any  curve  that  will  serve,  the  corresponding  curve 
may  be  found.  There  would  be,  therefore,  almost  an  infinite 
number  of  curves,  that  would  fulfill  the  requirements  of  correct 
tooth  outlines.  But  in  practice  two  kinds  of  curves  are  found  so 
convenient  that  they  are  most  commonly,  though  not  exclusively, 
used.     They  are  cycloidal  and  involute  curves. 

42.  It  is  assumed  that  the  character  of  cycloidal  curves  and 
method  of  drawing  them  is  understood. 

In  Fig.  40,  let  b  and  c  be  the  pitch  circles  of  a  pair  of  wheels, 
always  in  contact  at  be.  Also  let  m  be  the  describing  circle  in  con- 
tact with  both  at  the  same  point.  M  is  the  describing  point. 
When  one  curve  rolls  upon  another,  the  centro  of  their  relative 
motion  is  always  their  point  of  contact.  For,  since  the  motion  of 
rolling  excludes  slipping,  the  two  bodies  must  be  stationary,  relg,- 
tively  to  each  other,  at  their  point  of  contact ;  and  bodies  that 
move  relatively  to  each  other  can  have  but  one  such  stationary 
point  in  common  —  their  centro.     When,  therefore,  m  rolls  in  or 


^^  or  thb"^ 

IVBRSI 


I 


TOOTHED  WHEELS,    OR   GEARS.  43 


upon  b  or  c,  its  centro  relatively  to  either  is  their  point  of  contact. 
The  point  M,  therefore,  must  descrihe  curves  whose  direction  at 
any  point  is  at  right  angles  to  a  line  joining  that  point  to  the  point 
of  contact  of  m  with  the  circle.  Suppose  the  two  circles  b  and  c 
to  revolve  about  their  centres,  being  always  in  contact  at  be  ;  sup- 
pose m  to  rotate  at  the  same  time,  the  three  circles  being  always  in 
contact  at  one  point.  The  point  M  will  then  describe  simultane- 
ously a  curve,  b',  on  the  plane  of  b,  and  a  curve,  c\  on  the  plane  of  c. 
Since  M  describes  the  curves  simultaneously,  it  will  always  be  the 
point  of  contact  between  them  in  any  position.  And  since  the 
point  M  moves  always  at  right  angles  to  a  line  which  joins  it  to  be, 
therefore  the  normal  to  the  tooth  surfaces  at  their  point  of  contact 
will  always  pass  through  be,  and  the  condition  for  constant  velocity 
ratio  transmission  is  fulfilled.  But  these  curves  are  precisely  the 
epicycloid  and  hypocycloid  that  would  be  drawn  by  the  point  M  in 
the  generating  circle,  by  rolling  on  the  outside  of  b  and  inside  of  c. 
Obviously,  then,  the  epicycloids  and  hypocycloids  generated  in  this 
w^ay,  used  as  tooth  profiles,  will  transmit  a  constant  velocity  ratio. 

This  proof  is  independent  of  the  size  of  the  generating  circle,  and 
its  diameter  may  therefore  equal  the  radius  of  b.  Thein  the  hypo- 
cycloids  generated  by  rolling  within  b  would  be  straight  lines  coin- 
ciding with  the  radius  of  b.  In  this  case  the  profiles  of  the  teeth  of 
h  become  radial  lines  ;  and  therefore  the  teeth  are  thinner  at  the 
base  than  at  the  pitch  line  ;  for  this  reason  they  are  weaker  than  if 
a  smaller  generating  circle  had  been  used.  All  tooth  curves  gener- 
ated with  the  same  generating  circle  will  work  together,  the  pitch 
being  the  same.  It  is  therefore  necessary  to  use  the  same  generat- 
ing circle  for  a  set  of  gears  which  need  to  interchange. 

The  describing  circle  may  be  made  still  larger.  In  the  first  case 
the  curves  described  have  their  convexity  in  the  same  direction, 
i.  e.,  they  lie  on  the  same  side  of  a  common  tangent.  When  the 
diameter  of  the  describing  circle  is  made  equal  to  the  radius  of  b, 
one  curve  becomes  a  straight  line  tangent  to  the  other  curve.  As 
the  describing  circle  becomes  still  larger,  the  curves  have  their 
convexity    in    opposite    directions.     As    the    circle   approximates 


44  MACHINE    DESIGN. 

equality  with  b,  the  hypocycloid  grows  shorter,  and  finally,  when 
the  describing  circle  equals  b,  it  becomes  a  point  which  is  the  gener- 
ating point  in  b,  which  is  now  the  generating  circle.  If  this  point 
could  be  replaced  by  a  pin  having  no  sensible  diameter,  it  would 
engage  with  the  epicycloid  generated  by  it  in  the  other  gear  to 
transmit  a  constant  velocity  ratio.  But  a  pin  without  sensible 
diameter  will  not  serve  as  a  wheel  tooth,  and  a  proper  diameter 
must  be  assumed,  and  a  new  curve  laid  off  to  engage  with  it  in  the 
other  gear.  In  Fig.  41,  AB  is  the  epicycloid  generated  by  a  point 
in  the  circumference  of  the  other  pitch  circle.  CD  is  the  new  curve 
drawn  tangent  to  a  series  of  positions  of  the  pin  as  shown.  The 
pin  will  engage  with  this  curve,  CE,  and  transmit  the  constant 
velocity  ratio  as  required.  In  Fig.  40,  let  it  be  supposed  that  when 
the  three  circles  rotate  constantly  tangent  to  each  other  at  the  pitch 
point  be,  a  pencil  is  fastened  at  the  point  M  in  the  circumference  of 
the  describing  circle.  If  this  pencil  be  supposed  to  mark  simul- 
taneously upon  the  planes  of  b,  c,  and  that  of  the  paper,  it  will 
describe  upon  b  an  epicycloid,  on  c  a  hypocycloid,  and  on  the  plane 
of  the  paper  an  arc  of  the  describing  circle.  Since  M  is  always 
the  point  of  contact  of  the  cycloidal  curves  (because  it  generates 
them  simultaneously),  therefore,  in  cycloidal  gear  teeth,  the  Iocuh 
or  path  of  the  point  of  contact  is  an  arc  of  the  describing  circle. 

43.  In  the  cases  already  considered,  where  an  epicycloid  in  one 
wheel  engages  with  a  hypocycloid  in  the  other,  the  contact  of  the 
teeth  with  each  other  is  all  on  one  side  of  the  line  of  centres.  Thus, 
in  Fig.  40,  if  the  motion  be  reversed,  the  curves  will  be  in  contact 
until  M  returns  to  be  along  the  arc  MD-bc  ;  but  after  M  passes  be 
contact  will  cease.  If  c  were  the  driving  wheel,  the  point  of  con- 
tact would  approach  the  line  of  centres  ;  if  b  were  the  driving  wheel 
the  point  of  contact  would  recede  from  the  line  of  centres.  Exper- 
ience shows  that  the  latter  gives  smoother  running  because  of  better 
conditions  as  regards  friction  between  the  tooth  surfaces.  It  would 
be  desirable,  therefore,  that  the  wheel  with  the  epicycloidal  curves 
should  always  be  the  driver.  But  it  should  be  possible  to  use  either 
wheel  as  driver  to  meet  the  varying  conditions  in  practice. 


TOOTHED    WHEELS,    OK    GEARS.  45 

Another  reason  why  contact  should  not  be  all  on  one  side  of  the 
line  of  centres  may  be  explained  as  follows  : 

Definitions.  —  The  angle  through  which  a  gear  wheel  turns,  while 
one  of  its  teeth  is  in  contact  with  the  corresponding  tooth  in  the 
other  gear,  is  called  the  angle  of  action.  The  arc  of  the  pitch  circle 
corresponding  to  the  angle  of  action  is  called  the  arc  of  action. 

The  arc  of  action  must  be  greater  than  the  "pitch  arc"  (the  arc 
of  the  pitch  circle  that  includes  one  tooth  and  one  space),  or  else 
contact  will  cease  between  one  pair  of  teeth  before  it  begins  between 
the  next  pair.  .  Constrain  men  t  would  therefore  not  be  complete. 

In  Fig.  42,  let  AB  and  CD  be  the  pitch  circles  of  a  pair  of  gears, 
and  E  the  describing  circle.  Let  an  arc  of  action  be  laid  off  on 
each  of  the  circles  from  P,  as  Pa,  Pc,  and  Pe.  Through  e,  about  the 
centre  0,  draw  an  addendum  circle,  i.  e.,  the  circle  which  limits  the 
points  of  the  teeth.  Since  the  circle  E  is  the  path  of  the  point  of 
contact,  and  since  the  addendum  circle  limits  the  points  of  the 
teeth,  their  intersection,  e,  is  the  point  at  which  contact  ceases, 
rotation  being  as  indicated  by  the  arrow.  If  the  pitch  arc  just 
equals  the  assumed  arc  of  action,  contact  will  be  just  beginning  at 
P  when  it  ceases  at  e  ;  but  if  the  pitch  arc  be  greater  than  the  arc 
of  action,  contact  will  not  begin  at  P  till  after  it  has  ceased  at  e, 
and  there  will  be  an  interval  when  AB  will  not  drive  CD.  The 
greater  the  arc  of  action  the  greater  the  distance  of  e  from  P  on 
the  circumference  of  the  describing  circle.  The  direction  of  pressure 
between  the  teeth  is  always  a  normal  to  the  tooth  surface,  and  this 
always  passes  through  the  pitch  point ;  therefore,  the  greater  the  arc  of 
action,  i.e.,  the  greater  the  distance  oie  from  P,  the  greater  the  obliquity 
of  the  line  of  pressure.  The  pressure  may  be  resolved  into  two  com- 
ponents, one  at  right  angles  to  the  line  of  centres,  and  the  other 
parallel  to  it.  The  first  is  resisted  by  the  teeth  of  the  follower 
wheel,  and  therefore  produces  rotation  ;  the  second  is  resisted  at  the 
journal,  and  produces  pressure,  with  resulting  friction.  Hence,  it 
follows  that  the  greater  the  arc  of  action,  the  greater  will  be  the 
average  obliquity  of  the  line  of  pressure,  and  therefore  the  greater 
the  component  of  the  pressure  that  produces  wasteful  friction.     If 


46  MACHINE   DESIGN. 

it  can  be  arranged  so  that  the  arc  of  action  shall  be  partly  on  each 
side  of  the  line  of  centres,  the  arc  of  action  may  be  made  greater 
than  the  pitch  arc  (usually  equal  to  about  1^  times  the  pitch 
arc);  then  the  obliquity  of  the  pressure  line  may  be  kept  within 
reasonable  limits,  contact  between  the  teeth  will  be  insured  in  all 
positions,  and  either  wheel  may  be  the  driver.  This  is  accomplished 
by  using  two  describing  circles  as  in  Fig.  43.  Suppose  the  four 
circles  A,  B,  «,  and  /5,  to  roll  constantly  tangent  at  P.  «  will  de- 
scribe an  epicycloid  on  the  plane  of  B,  and  a  hypocycloid  on  the 
plane  of  A.  These  curves  will  engage  with  each  other  to  drive 
correctly.  /5  will  describe  an  epicycloid  on  A,  and  a  hypocycloid 
on  B.  These  curves  will  engage  also,  to  drive  correctly.  If  the 
epi-  and  hypocycloid  in  each  gear  be  drawn  through  the  same  point 
on  the  pitch  circle,  a  double  curve  tooth  outline  will  be  located, 
and  one  curve  will  engage  on  one  side  of  the  line  of  centres,  and  the 
other  on  the  other  side.  If  A  drives  as  indicated  by  the  arrow, 
contact  will  begin  at  D,  and  the  point  of  contact  will  follow  an  arc 
of  a  to  P,  and  then  an  arc  of  /5  to  C. 

44.  Involute  Tooth  Outlines.  —  If  a  string  be  wound  around  a 
cylinder  and  a  pencil  point  attached  to  its  end,  this  point  will 
trace  an  involute,  as  the  string  is  unwound  from  the  cylinder.  Or, 
if  the  point  be  constrained  to  follow  a  tangent  to  the  cylinder,  and 
the  string  be  unwound  by  rotating  the  cylinder  about  its  axis,  the 
point  will  trace  an  involute  on  a  plane  that  rotates  with  the  cylin- 
der. Illustration.  —  Let  «,  Fig.  44,  be  a  circular  piece  of  wood,  free 
to  rotate  about  C  ;  /5  is  a  circular  piece  of  cardboard  made  fast  to 
a ;  AB  is  a  straight-edge  held  on  the  circumference  of  a^  having  a 
pencil  point  at  B.  As  B  moves  along  the  straight-edge  to  A,  «  and 
/5  rotate  about  C,  and  B  traces  an  involute  DB  upon  /5.  The  rela- 
tive motion  of  the  tracing  point  and  /?  being  just  the  same  as  if  the 
string  had  been  simply  unwound  from  a,  fixed.  If  the  tracing 
point  is  caused  to  return  along  the  straight-edge  it  will  trace  the 
involute  BD  in  a  reverse  direction. 

The  centro  of  the  tracing  point  is  always  the  point  of  tangency 
of  the  string  with  the  cylinder  ;  therefore  the  string,  or  straight- 


or  TBS 


TOOTHED   WHEELS,    OR   GEARS.  47 

edge,  in  Fig.  45,  is  always  at  right  angles  to  the  direction  of  motion 
of  the  tracing  point,  and  hence  is  always  a  normal  to  the  involute  curve. 
Let  «  and  /5,  Fig.  45,  be  two  base  cylinders  ;  let  AB  be  a  cord  wound 
upon  a  and  /5  and  passing  through  the  centro  P,  which  corresponds 
to  the  required  velocity  ratio.  Let  a  and  /?  be  supposed  to  rotate  so 
that  the  cord  is  wound  from  /?  upon  a.  Then  any  point  in  the  cord 
will  move  from  A  toward  B,  and,  if  it  be  a  tracing  point,  will  trace 
an  involute  of  /5  on  the  plane  of  /?  (extended  beyond  the  base  cylin- 
der), and  will  also  trace  an  involute  of  «  upon  the  plane  of  «. 
These  two  involutes  will  serve  for  tooth  profiles  for  the  transmission 
of  the  required  constant  velocity  ratio,  because  AB  is  the  constant 
normal  to  both  curves  at  their  point  of  contact,  and  it  passes 
through  P,  the  centro  corresponding  to  the  required  velocity  ratio. 
Hence,  the  necessary  condition  is  fulfilled. 

Since  a  point  in  the  line  ^5  describes  involute  curves  simul- 
taneously, the  point  of  contact  of  the  curves  is  always  in  the  line 
AB.     And  hence  ^^  is  the  path  of  the  point  of  contact. 

One  of  the  advantages  of  involute  curves  for  tooth  profiles  is 

that  a  change  in  distance  between  centres  of  the  gears,  does  not 

interfere  with  the  transmission  of  a  constant  velocity  ratio.     This 

may   be   proved    as   follows :  In   Fig.   45,   from   similar   triangles 

07?        OP 

■pzr-r  =  T^rn  i  that  is,  the  ratio  of  the  radii  of  the  base  circles  is 
OA       OP 

equal  to  the  ratio  of  the  radii  of  the  pitch  circles.  This  ratio 
equals  the  inverse  ratio  of  angular  velocities  of  the  gears.  Suppose 
now  that  0  and  0'  be  moved  nearer  together  :  the  pitch  circles  will 
be  smaller,  but  the  ratio  of  their  radii  must  be  equal  to  the  un- 
changed ratio  of  the  radii  of  the  base  circles,  and  therefore  the 
velocity  ratio  remains  unchanged.  Also  the  involute  curves,  since 
they  are  generated  from  the  same  base  cylinders,  will  be  the  same 
as  before,  and  therefore,  with  the  same  tooth  outlines,  the  same 
constant  velocity  ratio  will  be  transmitted  as  before. 

45.  Definitions.  —  If  the  pitch  circle  be  divided  into  as  many 
equal  parts  as  there  are  teeth  in  the  gear,  the  arc  included  between 
two  of  these  divisions  is  the  circular  pitch  of  the  gear.     Circular 


48  MACHINE    DESIGN. 

pitch  may  also  be  defined  as  the  distance  on  the  pitch  circle  occu- 
pied by  a  tooth  and  a  space  ;  or,  otherwise,  it  is  the  distance  on  the 
pitch  circle  from  any  point  of  a  tooth  to  the  corresponding  point  in 
the  next  tooth.  A  fractional  tooth  is  impossible,  and  therefore  the 
circular  pitch  must  be  such  a  value  that  the  y)itch  circumference  is 
divisible   by  it.     Let  P --circular    pitch  in  inches;  let  D  =  pitch 

diameter  in  inches  ;  N  =number  of  teeth  ;  then  NP^=7rD  ;  N=  — ; 

NP  ttD 

D  = ;  P  =  — — .     From    these    relations    any    one    of    the    three 

values,  P,  D,  and  N,  may  be  found  if  the  other  two  are  given. 

Diametral  pitch  is  the  number  of  teeth  per  inch  of  pitch  diameter. 

N 
Thus,  if  p=  diameter  pitch,  p=~p:'     Multiplying  the  two  expres- 

sions,  P= -^  and   P  =  t^j  together,    gives    Pp— -^.|y=^-     ^-^^? 

the  product  of  diametral  and  circular  pitch  =7r.  Circular  pitch  is 
usually  used  for  large  cast  gears,  and  for  mortice  gears  (gears  with 
wooden  teeth  inserted).  Diametral  pitch  is  usually  used  for  small 
cut  gears. 

In  Fig.  46,  h,  e,  and  k,  are  pitch  points  of  the  teeth  ;  ab  is  the 
face  of  the  tooth  ;  hm  is  the  flank  of  the  tooth  ;  AD  is  the  total  depth 
of  the  tooth  ;  AC  is  the  working  depth;  AB  is  the  addendum;  a 
circle  through  A  is  the  addendum  circle.  Clearance  is  the  excess  of 
total  depth  over  working  depth,  =  CD.  Backlash  is  the  width  of 
space  on  the  pitch  line,  minus  the  width  of  the  tooth  on  the  same 
line.  In  cast  gears  whose  tooth  surfaces  are  not  '' tooled"  backlash 
needs  to  be  allowed,  because  of  unavoidable  imperfections  in  the 
surfaces.  In  cut  gears,  however,  it  may  be  reduced  almost  to  zero, 
and  the  tooth  and  space,  measured  on  the  pitch  circle,  may  be  con- 
sidered equal. 

46.  Racks.  —  A  rack  is  a  wheel  whose  pitch  radius  is  infinite  ; 
its  pitch  circle,  therefore,  becomes  a  straight  line,  and  is  tangent 
to  the  pitch  circle  of  the  wheel,  or  pinion  with  which  the  rack 
engages.     The  line  of  centres  is  a  normal  to  the  pitch  line  of  the 


*      0»  TttB 


^ 


TOOTHED    WHEELS,    OR   GEARS.  49 

rack,  through  the  centre  of  the  pitch  circle  of  the  pinion.  The 
pitch  of  the  rack  is  determined  hj  laying  off  the  circular  pitch  of 
the  engaging  wheel  on  the  pitch  line  of  the  rack.  The  curves  of 
the  rack  teeth,  like  those  of  wheels  of  finite  radius,  may  he  generated 
hy  a  point  in  the  circumference  of  a  circle  which  rolls  on  the  pitch 
circle.  Since,  however,  the  pitch  circle  is  now  a  straight  line,  the 
tooth  curves  will  he  cycloids,  both  for  flanks  and  faces.  In  Fig.  47, 
AB  is  the  pitch  circle  of  the  pinion,  and  CD  is  the  pitch  line  of  the 
rack  ;  a  and  h  are  describing  circles.  Suppose,  as  before,  that  all 
move  without  slipping,  and  are  constantly  tangent  at  P.  A  point 
in  the  circumference  of  a  will  then  describe  simultaneously  a 
cycloid  on  CD,  and  a  hypocycloid  within  AB.  These  will  be 
correct  tooth  outlines.  Also,  a  point  in  the  circumference  of  h 
will  describe  a  cycloid  on  CD  and  an  epicycloid  on  AB.  These 
will  be  correct  tooth  outlines.  To  find  the  path  of  the  point 
of  contact,  draw  the  addendum  circle  EF  of  the  pinion,  and  the 
addendum  line  GIT  of  the  rack.  When  the  pinion  turns  clockwise 
and  drives  the  rack,  contact  will  begin  at  e  and  follow  arcs  of  the 
describing  circles  through  P  to  K.  It  is  obvious  that  a  rack  cannot 
be  used  where  rotation  is  continuous  in  one  direction,  but  only 
where  motion  is  reversed. 

Involute  curves  may  also  be  used  for  the  outlines  of  rack  teeth. 
Let  CD  and  CD',  Fig.  48,  be  the  pitch  lines.  When  it  is  required 
to  generate  involute  curves  for  tooth  outlines,  for  a  pair  of  gears  of 
finite  radius,  a  line  is  drawn  through  the  pitch  point  at  a  given 
angle  to  the  line  of  centres  (usually  75°);  this  line  is  the  path  of 
the  point  which  generates  two  involutes  simultaneously,  and  there- 
fore the  path  of  the  point  of  contact  between  the  tooth  curves.  It 
is  also  the  common  tangent  to  the  two  base  circles,  which  may  now 
be  drawn  and  used  for  the  describing  of  the  involutes.  To  apply 
this  to  the  case  of  a  rack  and  pinion,  draw  EF,  Fig.  48.  The  base 
circles  must  be  drawn  tangent  to  this  line  ;  AB  will  therefore  be 
the  base  circle  for  the  pinion.  But  the  base  circle  in  the  rack  has 
an  infinite  radius,  and  a  circle  of  infinite  radius  drawn  tangent  to 
EF  would  be  a  straight  line  coincident  with  EF.     Therefore  EF  is 


50  MACHINE    DESIGN. 

the  base  line  of  the  rack.  But  an  involute  to  a  base  circle  of 
infinite  radius  is  a  straight  line  normal  to  the  circumference  —  in 
this  case  a  straight  line  perpendicular  to  EF.  Therefore  the  tooth 
profiles  of  a  rack  in  the  involute  system  will  always  be  straight 
lines  perpendicular  to  the  path  of  the  describing  point,  and  passing 
through  the  pitch  points.  If,  in  Fig.  48,  the  pinion  move  clockwise 
and  drive  the  rack,  the  contact  will  begin  at  E,  the  intersection  of 
the  addendum  line  of  the  rack  GH  and  the  base  circle  AB  of  the 
pinion,  and  will  follow  the  line  EF  through  P  to  the  point  where 
EF  cuts  the  addendum  circle  TM  of  the  pinion. 

47.  Annular  Gears.  —  Both  centres  of  a  pair  of  gears  may  be  on 
the  same  side  of  the  pitch  point.  This  arrangement  corresponds  to 
what  is  known  as  an  annular  gear  and  pinion.  Thus,  in  Fig.  49, 
AB  and  CD  are  the  pitch  circles,  and  their  centres  are  both  above 
the  pitch  point  P.  Teeth  may  be  constructed  to  transmit  rotation 
between  AB  and  CD.  AB  will  be  an  ordinary  spur  pinion,  but  it 
is  obvious  that  CD  becomes  a  ring  of  metal  with  teeth  on  the  inside, 
/.  6.,  it  is  an  annular  gear.  In  this  case  «  and  /?  may  be  describing 
circles,  and  a  point  in  the  circumference  of  «  will  describe  hypo- 
cycloids  simultaneously  on  the  planes  of  AB  and  CD ;  and  a  point 
in  the  circumference  of  ft  will  describe  epicycloids  simultaneously  on 
the  planes  of  AB  and  CD.  These  will  engage  to  transmit  a  con- 
stant velocity  ratio.  Obviously  the  space  inside  of  an  annular  gear 
corresponds  to  a  spur  gear  of  the  same  pitch  and  pitch  diameter, 
with  tooth  curves  drawn  with  the  same  describing  circle.  Let  EF 
and  GH,  Fig.  49,  be  the  addendum  circles.  If  the  pinion  move 
clockwise,  driving  the  annular  gear,  the  path  of  the  point  of  contact 
will  be  from  e  along  the  circumference  of  «  to  P,  and  from  P  along 
the  circumference  of  ft  to  K. 

The  construction  of  involute  teeth  for  an  annular  gear  and 
pinion  involves  exactly  the  same  principle  as  in  the  case  of  a  pair 
of  spur  gears.  The  only  difference  of  detail  is  that  the  describ- 
ing point  is  in  the  tangent  to  the  base  circles  produced  instead  of 
being  between  the  points  of  tangency.  Let  0  and  0',  Fig.  50,  be 
the  centres,  and  AB  and  //  the  pitch  circles  of  an  annular  gear 


Z3 


C&£A 


FHD 


r/a.  4^. 


H 


r/(7.^o. 


£\ 


TOOTHED    WHEELS,    OR    GEARS.  51 

and  pinion.  Through  P^  the  point  of  tangency  of  the  pitch  circles, 
draw  the  path  of  the  point  of  contact,  at  the  given  angle  with  the 
line  of  centres.  With  0  and  ()'  as  centres  draw  tangent  circles  to  this 
line.  These  will  be  the  involute  base  circles.  Let  the  tangent  be 
replaced  by  a  cord,  made  fast  say  at  K' ,  winding  on  the  circumfer- 
ence of  the  base  circle  CK\  to  /),  and  then  around  the  base  circle 
FE  in  the  direction  of  the  arrow,  and  passing  over  the  pulley  G 
which  holds  it  in  line  with  PB.  If  rotation  be  supposed  to  occur 
with  the  two  pitch  circles  always  tangent  at  P  without  slipping, 
any  point  in  the  cord  beyond  P  toward  (r,  will  describe  an  involute 
on  the  plane  /-/,  and  another  on  the  plane  of  AB.  These  will  be 
the  correct  involute  tooth  profiles  required.  Draw  NQ  and  LM,  the 
addendum  circles.  Then  if  the  pinion  move  clockwise,  driving  the 
annular  gear,  the  point  of  contact  starts  from  e  and  moves  along 
the  line  GH  through  P  to  K. 

When  a  pair  of  spur  gears  mesh  with  each  other,  the  direction  of 
rotation  is  reversed.  But  an  annular  gear  and  pinion  meshing 
together,  rotate  in  the  same  direction. 

48.  Interchangeable  Sets  of  Gears.  —  In  practice  it  is  desirable  to 
have  "interchangeable  sets"  of  gears  ;  i.  e.,  sets  in  which  any  gear 
will  "mesh"  correctly  with  any  other,  from  the  smallest  pinion  to 
the  rack,  and  in  which,  except  for  limiting  conditions  of  size,  any 
spur  gear  will  mesh  with  any  annular  gear.  Interchangeable  sets 
may  be  made  in  either  the  cycloidal  or  involute  system.  A  neces- 
sary condition  in  any  set  is,  that  the  pitch  shall  be  constant ;  be- 
cause the  thickness  of  tooth  on  the  pitch  line  must  always  equal 
the  width  of  the  space  (less  clearance).  If  this  condition  is 
unfulfilled  they  cannot  engage,  whatever  the  form  of  the  tooth 
outlines. 

The  second  condition  for  an  interchangeable  set  in  the  cycloidal 
system  is  that  the  size  of  the  describing  circle  shall  be  constant.  If 
the  diameter  of  the  describing  circle  equal  the  radius  of  the  smallest 
pinion's  pitch  circle,  the  flanks  of  this  pinion's  teeth  will  be  radial 
lines,  and  the  tooth  will  therefore  be  thinner  at  the  base  than  at 
the  pitch  line.     As  the  gears  increase  in  size  with  this  constant  size 


52  MACHINE    DESIGN. 

of  describing  circle,  the  teeth  grow  thicker  at  the  base  ;  hence,  the 
weakest  teeth  are  those  of  the  smallest  pinion. 

It  is  found  unadvisable  to  make  a  pinion  with  less  than  twelve 
teeth.  If  the  radius  of  a  fifteen-tooth  pinion  be  selected  for  the 
diameter  of  the  describing  circle,  the  flanks  in  a  twelve-tooth  j)inion 
will  be  very  nearly  parallel,  and  may  therefore  be  cut  with  a  mill- 
ing cutter.  This  would  not  be  possible  if  the  describing  circle 
were  made  larger,  causing  the  space  to  become  wider  at  the  bot- 
tom than  at  the  pitch  circle.  Therefore  the  maximum  describ- 
ing circle  for  milled  gears  is  one  whose  diameter  equals  the 
pitch  radius  of  a  tifteen-tooth  pinion,  and  it  is  the  one  usually 
selected.  Each  change  in  the  number  of  teeth  with  constant  pitch 
causes  a  change  in  the  size  of  the  pitch  circle.  Hence,  the  form  of 
the  tooth  outline,  generated  by  a  describing  circle  of  constant 
diameter,  also  changes.  For  any  pitch,  therefore,  a  separate  cutter 
would  be  required  corresponding  to  every  number  of  teeth,  to  in- 
sure absolute  accuracy.  Practically,  however,  this  is  not  necessary. 
The  change  in  the  form  of  tooth  outline  is  much  greater  in  a  small 
gear,  for  any  increase  in  the  number  of  teeth,  than  in  a  large  one. 
It  is  found  that  twenty-four  cutters  will  cut  all  possible  gears  of 
any  pitch  with  sufficient  practical  accuracy.  The  range  of  these 
cutters  is  indicated  in  the  following  table,  taken  from  Brown  & 
Sharpe's  "Treatise  on  Gearing"  : 
Cutter  A  cuts  12  teeth. 


B 

u 

18 

a 

C 

il 

14 

u 

D 

li 

15 

li 

E 

a 

16 

li 

F 

li 

17 

u 

G 

11 

18 

li 

H 

it 

19 

li 

I 

a 

20 

li 

J 

u 

21  to  22  teeth 

K 

u 

28  to  24     " 

L 

a 

24  to  26     '^ 

Cutter  M  cuts 

i    27  to    29  teeth 

(( 

N 

80  to    SS     " 

li 

0 

84  to    87     " 

ii 

P 

88  to    42     " 

n 

Q 

48  to    49     " 

11 

R 

50  to    59     " 

'' 

S 

60  to    74     " 

u 

T 

75  to    99     " 

11 

U 

100  to  149     " 

li 

V 

150  to  249     " 

11 

w 

250  to  rack. 

ii 

X 

rack. 

TOOTHED   WHEELS,    OR    GEARS.  53 

These  same  principles  of  interchangeable  sets  of  gears,  with 
cycloidal  tooth  outlines,  apply  not  only  to  small  milled  gears  as 
above,  but  also  to  large  cast  gears  with  tooled  or  untooled  tooth 
surfaces. 

49.  Interchangeable  Involute  Gears.  —In  the  involute  system  the 
second  condition  of  interchangeability  is  that  the  angle  between  the 
common  tangent  to  the  base  circles  and  the  line  of  centres  shall  be  con- 
stant. This  may  be  shown  as  follows  :  Draw  the  line  of  centres, 
AB,  Fig.  51.  Through  P,  the  assumed  pitch  point,  draw  CD,  and 
let  it  be  the  constant  common  tangent  to  all  base  circles  from  which 
involute  tooth  curves  are  to  be  drawn.  Draw  any  pair  of  pitch 
circles  tangent  at  P,  with  their  centres  in  the  line  AB.  About 
these  centres  draw  circles  tangent  to  CD  ;  these  are  base  circles, 
and  CD  may  represent  a  cord  that  winds  from  one  upon  the  other. 
A  point  in  this  cord  will  generate,  simultaneously,  involutes  that 
will  engage  for  the  transmission  of  a  constant  velocity  ratio.  But 
this  is  true  of  any  pair  of  circles  that  have  their  centres  in  AB,  and 
are  tangent  to  CD.  Therefore,  if  the  pitch  is  constant,  any  pair  of 
gears  that  have  the  base  circles  tangent  to  the  line  CD,  will  mesh 
together  properl3^  As  in  the  cycloidal  gears,  the  involute  tooth 
curves  vary  with  a  variation  in  the  number  of  teeth,  and,  for  abso- 
lute theoretical  accuracy,  there  would  be  required  for  each  pitch  as 
many  cutters  as  there  are  gears  with  different  numbers  of  teeth. 
The  variation  is  least  at  the  pitch  line,  and  increases  with  the  dis- 
tance from  it.  The  involute  teeth  are  usually  used  for  the  finer 
pitches,  and  the  cycloidal  teeth  for  the  coarser  pitches  ;  and  since 
the  amount  that  the  tooth  surface  extends  beyond  the  pitch  line  in- 
creases with  the  pitch,  it  follows  that  the  variation  in  form  of  tooth 
curves  is  greater  in  the  coarse  pitch  cycloidal  gears  than  in  the  fine 
pitch  involute  gears.  For  this  reason,  with  involute  gears,  it  is 
only  necessary  to  use  eight  cutters  for  each  pitch.  The  range  is 
shown  in  the  following  table,  which  is  also  taken  from  Brown  & 
Hharpe's  "Treatise  on  Gearing"  : 


54 


2 

3 

4 

5 

6 

7 

8 

MACHINE    DESIGN. 

3ls  from 

135 

teeth  to  racks. 

4i 

55 

to  134  inc 

•lusive, 

U 

35 

to 

54 

a 

26 

to 

34 

u 

21 

to 

25 

i:< 

17 

to 

20 

(( 

14 

to 

16 

u 

12 

to 

13 

50.  Laying  Out  Gear  Teeth.  Exact  and  Approximate  Methods.  — 
There  is  ordinarily  no  reason  why  an  exact  method  for  laying  ont 
cycloidal  or  involute  curves  for  tooth  outlines  should  not  he  used, 
either  for  large  gears  or  gear  patterns,  or  in  making  drawings.  It  is 
required  to  lay  out  a  cycloidal  gear.  The  pitch,  and  diameters  of 
pitch,  and  describing  circle  are  given. — Draw  the  pitch  circle.  From 
a  piece  of  thin  wood  cut  out  a  template  to  fit  a  segment  of  the  pitch 
circle  from  the  inside,  as  A,  Fig.  52.  Cut  another  template  to  fit  a 
segment  of  the  pitch  circle  from  the  outside,  as  B.  Also  cut  a 
wooden  disc  whose  diameter  equals  that  of  the  given  describing 
circle,  and  fix  a  tracing  point  in  its  circumference.  Divide  the 
pitch  circle  into  parts  equal  to  the  given  circular  pitch.  Let  P  be 
one  of  the  pitch  points.  Locate  A  so  that  its  curved  edge  coincides 
with  the  pitch  circle  at  the  right  of  P.  Roll  the  describing  circle 
on  A,  without  slipping,  so  that  the  epicycloid  described  by  the 
tracing  point  shall  pass  through  P.  Next  place  B  so  that  its  curved 
edge  coincides  with  the  pitch  circle  at  the  left  of  P,  and  roll  the 
circle  on  the  inside  of  B,  without  slipping,  so  that  the  hypocycloid 
described  by  the  tracing  point  shall  pass  through  P.  Thus  the 
outline  of  one  tooth  is  drawn,  aPh.  Cut  a  wooden  template  to  fit 
the  tooth  curve,  and  make  it  fast  to  a  wooden  arm  free  to  rotate 
about  0,  making  the  edge  of  the  template  coincide  with  aPh.  It 
may  now  be  swung  successively  to  the  other  pitch  points,  and  the 
tooth  outline  may  be  drawn  by  the  template  edge.  This  gives  one 
side  of  all  of  the  teeth.  The  arm  may  now  be  turned  over  and  the 
other  sides  of  the  teeth  may  be  drawn  similarly. 


TOOTHED   WHEELS,    OR   GEARS.  55 

51.  It  is  required  to  lay  out  exact  involute  teeth.  The  pitch, 
pitch  circle  diameter,  and  angle  of  the  common  tangent  are  given. 
—  Draw  the  pitch  circle,  Fig.  58,  and  the  line  of  centres  AB. 
Through  the  pitch  point,  P,  draw  CD,  the  common  tangent  to  the 
hase  circles,  making  the  angle  /5  with  the  line  of  centres.  Draw  the 
hase  circle  ahout  0,  tangent  to  CD.  Cut  a  wooden  template  to  fit 
the  base  circle  from  the  inside,  as  EF;  wind  on  this  template  a 
fine  cord  carrying  a  pencil  at  its  end,  and  then  unwind  this,  allow- 
ing the  pencil  to  trace  an  involute  curve,  ah,  which  will  be  a  correct 
tooth  form.  Let  a  template,  cut  to  fit  this  involute,  be  attached  to 
an  arm  free  to  rotate  about  0,  and  the  tooth  outlines  may  be  drawn 
as  before.  The  bottom  of  the  spaces  between  the  teeth  may  fall 
within  the  base  circle,  in  which  case  the  involute  curves  are  extended 
inward  by  radial  lines.* 

52.  The  following  formulas  are  given  to  assist  in  the  practical 
proportioning  of  gears  : 

Let  D  =  pitch  diameter. 
"    Z)i=  outside  diameter. 

"    D.^=  diameter  of  a  circle  through  the  bottom  of  spaces. 
''    P  ^  circular  pitch  =  space  on  the  pitch  circle  occupied  by  a 

tooth  and  a  space. 
"     p  =  diametral  pitch  =  number  of   teeth  per  inch  of   pitch 

circle  diameter. 
"    7V=  number  of  teeth. 
"      t  =  thickness  of  tooth  on  pitch  line. 
*'     a  =  addendum. 
*'     c  =  clearance. 
"     d  =  working  depth  of  spaces. 
"     (/i=  full  depth  of  spaces. 

*  Approximate  tooth  outlines  may  be  drawn  by  the  use  of  instruments, 
such  as  the  Willis  odontograph,  which  locates  the  centres  of  approximate  cir- 
cular arcs;  the  templet  odontograph,  invented  by  Prof.  S.  W.  Robinson;  or 
by  some  geometrical  or  tabular  method  for  the  location  of  the  centres  of  ap- 
proximate circular  arcs.  For  descriptions  see  "Elements  of  Mechanism," 
Willis;  ''Kinematics,"  McCord;  "Teeth  of  Gears,"  George  B.  Grant; 
"Treatise  on  Gearing,"  published  by  Brown  &  Sharpe. 


56  MACHINE    DESIGN. 

Then,  D,  =  ^L±l •   j)^  =  D  —  2(a  -{  c); 

TT  TT  Ft: 

P  =^  -Ti\  P  ^^  — 5   ^  =^  T.  "=  K- ^  ^^  backlash. 
P'  p  '  2       2/) 

J  r>  _  -J 

c  ~  Th ^=  oA  =  "ori'   d=  2a  ;  di=2a  -\r  c  ;  a  =  —  inches. 
10       20      p2\j  P 

The  following  dimensions  are  given  as  a  guide ;  they  may  be 
varied  as  conditions  of  design  require  :  Width  of  face  =  about  8P ; 
thickness  of  rim  =  1.25  f;  thickness  of  arms  =  1.25«;  no  taper. 
The  rim  may  be  reinforced  by  a  rib,  as  shown  in  Fig.  54.  Diame- 
ter of  hub  =^  2  X  diameter  of  shaft.  Length  of  hub  =  width  of 
face  4"  i"  ;  width  of  arm  at  junction  with  hub  =  i  circumference 
of  the  hub,  for  six  arms.  Make  arms  taper  about  |"  per  foot  on 
each  side. 

53,  Strength  of  Gear  Teeth.  —  The  maximum  work  transmitted 
by  a  shaft  per  unit  time  may  usually  be  accurately  estimated  ;  and, 
if  the  rate  of  rotation  is  known,  the  torsional  moment  may  be  found. 
Let  0,  Fig.  55,  represent  the  axis  of  a  shaft  perpendicular  to  the 
paper.  Let  A  =  maximum  work  to  be  transmitted  per  minute  : 
let  N  ==  revolutions  per  minute  ;  let  Fr  =  torsional  moment.  Then 
F  is  the  force  factor  of  the  work  transmitted,  and  27trN  is  the  space 
factor  of  the  work  transmitted.     Hence,  2Fr.rN  =  A,  and  Fr  =  tor- 

sional  moment  =  ;^-^r- 
2t:N 

If  the  work  is  to  be  transmitted  to  another  shaft  by  means  of  a 

spur  gear  whose  radius  is  r,,  then  for  equilibrium  F^r^  =  Fr,  and 

Fr 

F^  =  — .     F,  is  the  force  at  the  pitch   surface  of  the  gear  whose 

^1  • 

radius  is  r^,  i.  e.,  it  is  the  force  to  be  sustained  by  the  gear  teeth. 
Hence,  in  general,  the  force  sustained  by  the  teeth  of  a  gear  equals 
the  torsional  moment  divided  by  the  pitch  radius  of  the  gear. 


<r^ 

K 

//'^^/7^. 

I^\^ 

^      ' 

/V^^       ^"^><S'^ 

cl               \ 

0 

\\/y.^e.   J 

/^ 

^^ 

If 

^ 

5^/  /y.^^. 

'UKIVBRSITTj 


TOOTHED   WHEELS,    OR   GEARS.  57 

When  the  maximum  force  to  be  sustained  is  known  the  teeth 
may  be  given  proper  proportions.  The  dimensions  upon  which  the 
tooth  depends  for  strength  are  :  Thickness  of  tooth  =  t  ;  width  of 
face  of  gear  =  h  ;  and  depth  of  space  between  teeth  =^  I.  These  all 
become  known  when  the  pitch  is  known,  because  t  is  fixed  for  any 
pitch,  and  I  and  b  have  values  dictated  by  good  practice.  The 
value  of  b  may  be  varied  through  quite  a  range  to  meet  the  require- 
ments of  any  special  case. 

54.  In  the  design  the  tooth  will  be  treated  as  a  cantilever  with  a 

load  applied  at  its  end.     It  is  assumed  that  one  tooth  sustains  the 

entire  load  ;  i.  e.,  that  there  is  contact  only  between   one  pair  of 

teeth.     This  would  be  nearly  true  for  gears  with  low  numbers  of 

teeth  ;  but  in  high  numbered  gears  the  force  would  be  distributed 

over  several  pairs,  and  hence  they  would  have  an  excess  of  strength. 

It  is  also  assumed  that  the  tooth  has  the  same  thickness  from  the 

pitch  circle  to  its  root.     This  is  also  nearly  true  for  low  numbered 

gears,  while  high  numbered  gears  would  have  excess  of  strength  as 

a  result  of  this  assumption.     In  Fig.  56  let  P  =  force  at  the  pitch 

surface  of  the  gear  to  be  designed,  b  =  width  of  face,  I  =^  depth  of 

space,  and  d  =  thickness  of  the  tooth  at  the  pitch   circle.     From 

Mechanics  of  Materials  it  is  known  that  the  moment  of  flexure, 

SI 
PL  =       ;  in  which  S  is  the  unit  stress  in  the  outer  fibre  ;  /  is  the 
c 

b(P 
moment  of  inertia  of  the  cross  section,  =  rr^;   and  c  is  the  distance 

from  the  neutral  axis  to  the  outer  fibre,  =  ^.      Hence,     PI  =  —77—; 

M  U 

6Pl 

S  =  ,~.  Assume  a  value  for  diametral  pitch,  find  the  correspond- 
ing values  of  ft,  d,  and  I  from  table  on  page  58,  and  substitute  in  the 
above  equation.  S  now  becomes  known,  and  may  be  compared 
with  the  ultimate  strength  of  the  material  of  the  gear,  =  S^.     If  the 

S 
factor  of  safety,  =  -^,  is  a  proper  value,  the  assumed  pitch  is  right. 

If  not,  another  pitch  may  be  assumed  and  checked  as  before. 


58 


MACHINE    DESIGN. 


Table  I.  —  For  Use  ix  Designing  Gears. 


Diametral 
Pitch 

Circular 
Pitch 

Thickness 

of  Tooth 

on  the 

Width 
of  Face 

Safe  Stress 

for  Cast 
Iron  Gear 

Safe  Unit 

Stress  for 

Cast  Iron 

Gear 

Safe  Stress 

for  Cast 
Steel  Gear 

Safe  Unit 

Stress  for 

Cast  Steel 

Gears 

Pitch  Une 
d 

b 

Factor  of 
Safety=^/o 

Factor  of 
Safety -=io 

Factor  of 
Safety=6 

Factor  of 
Safety=6 

K 

6.283 

3.141 

20 

15250 

763 

61000 

3033 

M 

4.189 

2.094 

13 

6590 

507 

26390 

2030 

1 

3.141 

1.571 

9 

3442 

382 

13770 

1530 

m 

2.513 

1.256 

7K 

2250 

305 

9000 

1220 

i>t 

2.094 

1.047 

6 

1530 

255 

6120 

1020 

1^ 

1.795 

.897 

5K 

1200 

218 

4800 

872 

2 

1.571 

.785 

4>^ 

862 

192 

3450 

767 

^K 

1.396 

.698 

4 

684 

170 

2738 

682 

2% 

1.256 

.628 

3K 

532 

152 

2130 

610 

2% 

1.142 

.571 

3 

417 

139 

1668 

556 

3 

1.047 

.523 

2% 

318 

127 

1272 

509 

3K 

.897 

.449 

2 

216 

108 

864 

437 

4 

.785 

.393 

1% 
1% 

168 

9^ 

672 

383 

5 

.628 

.314 

124 

76 

498 

306 

6 

.523 

.262 

1% 

96 

63 

384 

253 

7 

449 

.224 

1% 

74 

54 

296 

216 

8 

.393 

.196 

IK 

60 

48 

240 

192 

9 

.349 

.174 

IM 

48 

42 

191 

170 

10 

.314 

.157 

lh6 

41 

38 

163 

153 

12 

.262 

.131 

»5i6 

30 

32 

120 

128 

14 

224 

.112 

13i6 

22 

27 

88 

109 

55.  The  work  of  approximation  may  be  avoided  by  the  use  of 
Table  I.  Column  1  gives  values  of  diametral  pitch.  Column  2 
gives  values  of  circular  pitch.  Column  3  gives  values  of  d. 
Column  4  gives  values  of  width  of  face,  =  ft,  corresponding  to 
good  practice.  If  this  value  of  h  is  accepted,  the  table  may  be 
used  as  follows :  The  maximum  working  force  at  the  pitch 
surface  of  the  gear  is  found  as  above.  If  this  value  is  found  in 
column  5,  the  value  of  diametral  pitch  horizontally  opposite  in 
column  1  may  be  used  for  a  cast  iron  gear,  with  a  factor  of  safety  of 
10.  If  the  value  is  not  found  in  column  5,  take  the  next  greater 
value,  and  use  the  corresponding  pitch.  This  will  slightly  increase 
the  factor  of  safety. 

If  the  conditions  of  the  design  require  some  different  value  for  ft, 
divide  the  maximum  working  force  at  the  pitch   surface   by  the 


^^^-^^ 


Of  THl 


^SIVBHSITT 


oar 


^£IP0 


TOOTHED   WHEELS,    OR   GEARS.  59 

width  of  face,  find  this  value,  or  the  next  greater,  in  column  6,  and 
use  the  corresponding  value  of  pitch. 

If  the  pitch  thus  determined  is  too  large  for  the  design,  a  steel 
casting  may  be  used,  and  the  pitch  will  be  determined  by  use  of 
column  7  or  8  as  above. 

56.  Non-Circular  Wheels.  —  Only  circular  centrodes  or  pitch 
curves  correspond  to  a  constant  velocity  ratio  ;  and  by  making  the 
pitch  curves  of  proper  form,  almost  any  variation  in  the  velocity 
ratio  may  be  produced.  Thus  a  gear  whose  pitch  curve  is  an  ellipse, 
rotating  about  one  of  its  foci,  may  engage  with  another  elliptical 
gear,  and  if  the  driver  has  a  constant  angular  velocity,  the  follower 
will  have  a  constantly  varying  angular  velocity.  If  the  follower  is 
rigidly  attached  to  the  crank  of  a  slider  crank  chain,  the  slider  will 
have  a  quick  return  motion.  This  is  sometimes  used  for  shapers 
and  slotting  machines.  When  more  than  one  fluctuation  of  veloc- 
ity per  revolution  is  required,  it  may  be  obtained  by  means  of 
"  lobed  gears "  ;  i.  e.,  gears  in  which  the  curvature  of  the  pitch 
curve  is  several  times  reversed.  If  a  describing  circle  be  rolled  on 
these  non-circular  pitch  curves,  the  tooth  outlines  will  vary  in 
different  parts  ;  hence,  in  order  to  cut  such  gears,  many  cutters 
would  be  required  for  each  gear.  Practically,  this  would  be  too 
expensive  ;  and  when  such  gears  are  used  the  pattern  is  accurately 
made,  and  the  cast  gears  are  used  without  "tooling"  the  tooth 
surfaces. 

57.  Bevel  Gears.  —  All  transverse  sections  of  spur  gears  are  the 
same,  and  their  axes  intersect  at  infinity.  Spur  gears  serve  to 
transmit  motion  between  parallel  shafts.  It  is  necessary  also  to 
transmit  motion  between  shafts  whose  axes  intersect.  In  this  case 
the  pitch  cylinders  become  pitch  cones  ;  the  teeth  are  formed  upon 
these  conical  surfaces,  the  resulting  gears  being  called  bevel  gears. 
To  illustrate,  let  a  and  6,  Fig.  57,  be  the  axes  between  which  the 
motion  is  to  be  transmitted  with  a  given  velocity  ratio.  This  ratio 
is  equal  to  the  ratio  of  the  length  of  the  line  A  to  that  of  B.  Draw 
a  line  CD  parallel  to  a,  at  a  distance  from  it  equal  to  the  length  of 


60  MACHINE   DESIGN. 

the  line  A.  Also  draw  the  line  CE  parallel  to  h,  at  a  distance  from 
it  equal  to  the  length  of  the  line  B.  Join  the  point  of  intersection 
of  these  lines  to  the  point  0,  the  intersection  of  the  given  axes. 
This  locates  the  line  CF,  which  is  the  line  of  contact  of  two  pitch 
cones  which  will  roll  together  to  transmit  the  required  velocity  ratio. 

For  —  =  t;?  and  if  it  be  supposed  that  there  are  frusta  of  cones  so 
ncB  ^  ^ 

thin  that  they  may  be  considered  cylinders,  their  radii  being  equal 

to  mc  and  nc,  it  follows  that  they  would  roll  together,  if  slipping  be 

prevented,  to  transmit  the  required  velocity  ratio.     But  all  pairs  of 

tHjC 
radii  of  these  pitch  cones  have  the  same  ratio,  =  — ,  and  therefore 

nc 

any  pair  of  frusta  of  the  pitch  cones  may  be  used  to  roll  together 
for  the  transmission  of  the  required  velocity  ratio.  To  insure  this 
result,  slipping  must  be  prevented,  and  hence  teeth  are  formed 
upon  the  selected  frusta  of  the  pitch  cones.  The  theoretical  deter- 
mination of  these  may  be  explained  as  follows  : 

Ist.  Cydoidal  Teeth.  —  If  a  cone  A  (Fig.  58),  be  rolled  upon 
another  cone,  B,  an  element  be  of  the  cone  A  will  generate  a  conical 
surface,  and  a  spherical  section  of  this  surface,  adb,  is  called  a 
spherical  epicycloid.  Also  if  a  cone,  A  (Fig.  59),  roll  on  the  inside 
of  another  cone,  0,  an  element  he  of  A  will  generate  a  conical  sur- 
face, a  spherical  section  of  which,  hda,  is  called  a  spherical  hypo- 
cycloid.  If  now  the  three  cones,  B,  C,  and  A,  roll  together,  always 
tangent  to  each  other  on  one  line,  as  the  cylinders  were  in  the  case 
of  spur  gears,  there  will  be  two  conical  surfaces  generated  by  an 
element  of  A  ;  one  upon  the  cone  B,  and  another  upon  the  cone  C. 
These  may  be  used  for  tooth  surfaces  to  transmit  the  required  con- 
stant velocity  ratio.  Because,  since  the  line  of  contact  of  the  cones 
is  the  axo  *  of  the  relative  motion  of  the  cones,  it  follows  that  a 
plane  normal  to  the  motion  of  the  describing  element  of  the  gen- 
erating cone  at  any  time,  will  pass  through  this  axo.  And  also, 
since  the  describing  element  is  always  the  line  of  contact  between 

*  An  axo  is  an  instantaneous  axis,  of  which  a  centre  is  a  projection. 


TOOTHED   WHEELS,    OR   GEARS.  61 

the  generated  tooth  surfaces,  the  normal  plane  to  the  line  of  contact 
of  the  tooth  surfaces  always  passes  through  the  axo,  and  the  con- 
dition of  rotation  with  a  constant  velocity  ratio  is  fulfilled. 

2d.  Involute  Teeth.  —  If  two  pitch  cones  are  in  contact  along 
an  element,  a  plane  may  he  passed  through  this  element,  making 
an  angle  (say  75°)  with  the  plane  of  the  axes  of  the  cones.  Tan- 
gent to  this  plane  there  may  l)e  two  cones,  whose  axes  coincide 
with  the  axes  of  the  pitch  cones.  If  a  plane  be  supposed  to  wind 
off  from  one  base  cone  upon  the  other,  the  line  of  tangency  of  the 
plane  with  one  cone  will  leave  the  cone  and  advance  in  the  plane 
toward  the  other  cone,  and  will  generate  simultaneously  upon  the 
pitch  cones  conical  surfaces,  and  spherical  sections  of  these  surfaces 
will  be  spherical  involutes.  These  surfaces  may  be  used  for  tooth 
surfaces,  and  will  transmit  the  required  constant  velocity  ratio,  be- 
cause the  tangent  plane  is  the  constant  normal  to  the  tooth  sur- 
faces at  their  line  of  contact,  and  this  plane  passes  through  the  axo 
of  the  pitch  cones. 

To  determine  the  tooth  surfaces  with  perfect  accuracy,  it  would 
be  necessary  to  draw  the  required  curves  on  a  spherical  surface, 
and  then  to  join  all  points  of  these  curves  to  the  point  of  intersec- 
tion of  the  axes  of  the  pitch  cones.  Practically  this  would  be  im- 
possible, and  an  approximate  method  is  used. 

If  the  frusta  of  pitch  cones  l>e  given,  B  and  C,  Fig.  60,  then 
points  in  the  base  circles  of  the  cones,  as  X,  M,  and  K,  will  move 
always  in  the  surface  of  a  sphere  whose  projection  is  the  circle 
LA  KM.  Properly,  the  tooth  curves  should  be  laid  out  on  the  sur- 
face of  this  sphere,  and  joined  to  the  centre  of  the  sphere  to  gen- 
erate the  tooth  surfaces.  Draw  cones  LGM  and  MHK  tangent  to 
the  sphere  on  circles  represented  in  projection  by  lines  LM  and  MK. 
If  now  tooth  curves  be  drawn  on  these  cones,  with  the  base  circle  of 
the  cones  as  pitch  circles,  they  will  very  closely  approximate  the 
tooth  curves  that  should  be  drawn  on  the  spherical  surface.  But  a 
cone  may  be  cut  along  one  of  its  elements  and  rolled  out,  or  de- 
veloped, upon  a  plane.  Let  MDH  be  a  part  of  the  cone  MHK,  de- 
veloped, and  let  MNG  be  a  part  of  the  cone  MGL,  developed.     The 


62  MACHINE   DESIGN. 

circular  arcs  MD  and  MN  may  be  used  just  as  pitch  circles  are  in 
the  case  of  spur  gears,  and  the  teeth  may  be  laid  out  in  exactly  the 
same  way,  the  curves  being  either  cycloidal  or  involute,  as  required. 
Then  the  developed  cones  may  be  wrapped  back,  and  the  curves 
drawn  may  serve  as  directrices  for  the  tooth  surfaces,  all  of  whose 
elements  converge  to  the  centre  of  the  sphere  of  motion. 

58.  The  teeth  of  spur  gears  may  be  cut  by  means  of  milling  cut- 
ters, because  all  transverse  sections  are  alike ;  but  with  bevel  gears 
the  conditions  are  different.  The  tooth  surfaces  are  conical  sur- 
faces, and  therefore  the  curvature  varies  constantly  from  one  end  of 
the  tooth  to  the  other.  Also  the  thickness  of  the  tooth  and  the 
width  of  space  vary  constantly  from  one  end  to  the  other.  But  the 
curvature  and  thickness  of  a  milling  cutter  cannot  vary,  and  there- 
fore a  milling  cutter  cannot  cut  an  accurate  bevel  gear.  Small 
bevel  gears  are,  however,  cut  with  milling  cutters  with  sufficient 
accuracy  for  practical  purposes.  The  cutter  is  made  as  thick  as 
the  narrowest  part  of  the  space  between  the  teeth,  and  its  curvature 
is  made  that  of  the  middle  of  the  tooth.  Two  cuts  are  made  for 
each  space.  Let  Fig.  61  represent  a  section  of  the  cutter.  For  the 
first  cut  it  is  set  relatively  to  the  gear  blank,  so  that  the  pitch  point 
a  of  the  cutter  travels  toward  the  apex  of  the  pitch  cone,  and  for 
the  second  cut  so  that  the  pitch  point  b  travels  toward  the  apex  of 
the  pitch  cone.  This  method  gives  an  approximation  to  the  re- 
quired form.  Gears  cut  in  this  manner  usually  need  to  be  filed 
slightly  before  they  work  satisfactorily.  Bevel  gears  with  abso- 
lutely correct  tooth  surfaces  may  be  made  by  planing.  Suppose  a 
planer  in  which  the  tool  point  travels  always  in  some  line  through 
the  apex  of  the  pitch  cone.  Then  suppose  that  as  it  is  slowly  fed 
down  the  tooth  surface,  it  is  guided  along  the  required  tooth  curve 
by  means  of  a  templet.  From  what  has  preceded  it  will  be  clear 
that  the  tooth  so  formed  will  be  correct.  Planers  embodying  these 
principles  have  been  designed  and  constructed  by  Mr.  Corliss  of 
Providence,  and  Mr.  Gleason  of  Rochester,  with  the  most  satisfac- 
tory results. 

59.  Design  of  Bevel  Gears.  —  Given  energy  to  be  transmitted,  rate 


► 


:^^-^^ 


OT  THX 


[UHIVBRSITT] 


o» 


-IPOIt 


TOOTHED   WHEELS,    OR   GEARS.  63 

of  rotation  of  one  shaft,  velocity  ratio,  and  angle  between  axes  ;  to 
design  a  pair  of  bevel  gears.  Locate  the  intersection  of  axes,  0, 
Fig.  62.  Draw  the  axes  OA  and  OB,  making  the  required  angle 
with  each  other.  Locate  OC,  the  line  of  tangency  of  the  pitch 
cones,  by  the  method  given  on  page  59.  Any  pair  of  frusta  of  the 
pitch  cones  may  be  selected  upon  which  to  form  the  teeth.  Special 
conditions  of  the  problem  usually  dictate  this  selection  approx- 
imately. Suppose  that  the  inner  limit  of  the  teeth  may  be  con- 
veniently at  D.  Then  make  DP,  the  width  of  face,  =  DO  ~  2.  Or, 
if  Pis  located  by  some  limiting  condition,  lay  off  PD  =  PO  -^  S. 
In  either  case  the  limits  of  the  teeth  are  defined  tentatively.  Now 
from  the  energy  and  the  number  of  revolutions  of  one  shaft  (either 
shaft  may  be  used),  the  moment  of  torsion  may  be  found.  The 
mean  force  at  the  pitch  surface  =  this  torsional  moment  -^  the 
mean  radius  of  the  gear  ;  i.  e.,  the  radius  of  the  point  M,  Fig.  62, 
midway  between  P  and  D.  The  pitch  corresponding  to  this  force 
may  be  found  from  Table  1.  This  would  be  the  mean  pitch  of  the 
gears.  But  the  pitch  of  bevel  gears  is  measured  at  the  large  end, 
and  diametral  pitch  varies  inversely  as  the  distance  from  0.  In 
this  case  the  distances  of  M  and  P  from  0  are  to  each  other  as  5  is 
to  6.     Hence  the  value  of  diametral  pitch  found  from  the  table 

X  ^  =  the  diametral  pitch  of  the  gear.  If  this  value  does  not  cor- 
respond with  any  of  the  usual  values  of  diametral  pitch,  the  rext 
smaller  value  may  be  used.  This  would  result  in  a  slightly  in- 
creased factor  of  safety.  If  the  diametral  pitch  thus  found,  multiplied 
by  the  diameter  corresponding  to  the  point  P,  does  not  give  an  integer 
for  the  number  of  teeth,  the  point  P  may  be  moved  outward  along 
the  line  OC,  until  the  number  of  teeth  becomes  an  integer.  This 
also  would  result  in  slight  increase  of  the  factor  of  safety.  The 
point  P  is  thus  finally  located,  the  corrected  width  of  face  =  P0-^-3, 
and  the  pitch  is  known.  The  drawing  of  the  gears  may  be  com- 
pleted as  follows  :  Draw  AB  perpendicular  to  PO.  With  A  and  B 
as  centres,  draw  the  arcs  PE  and  PF.  Use  these  as  pitch  arcs,  and 
draw  the  outlines  of  two  or  three  teeth  upon  them,  with  cycloidal 


64  MACHINE    DESIGN. 

or  involute  curves  as  required.  These  will  serve  to  show  the  form 
of  tooth  outlines.  From  P  each  way  along  the  line  AB  lay  off  the 
addendum  and  the  clearance.  From  the  four  points  thus  located 
draw  lines  toward  0,  terminating  in  the  line  DG.  The  tops  of  teeth 
and  the  bottoms  of  spaces  are  thus  defined.  Layoff  upon  AB  below 
the  bottoms  of  the  spaces,  a  space  about  equal  to  the  thickness  of 
the  tooth  on  the  pitch  circle.  This  gives  a  ring  of  metal  to  support 
the  teeth.  Join  this  to  a  properly  proportioned  hub  as  shown. 
The  plan  and  elevation  of  each  gear  may  now  be  drawn  by  the 
ordinary  methods  of  projection. 

60.  Skew  Bevel  Gears.  —  Spur  gears  serve  to  communicate  motion 
between  parallel  axes,  and  bevel  gears  between  axes  that  intersect. 
But  it  is  sometimes  necessary  to  communicate  motion  between  axes 
that  are  neither  parallel  nor  intersecting.  If  the  parallel  axes  are 
turned  out  of  parallelism,  or  if  intersecting  axes  are  moved  into 
different  planes,  so  that  they  no  longer  intersect,  the  pitch  surfaces 
become  hyperbolic  paraboloids  in  contact  with  each  other  along  a 
straight  line,  which  is  the  generatrix  of  the  pitch  surfaces.  These 
hyperbolic  paraboloids  rolled  upon  each  other,  circumferential 
slipping  being  prevented,  will  transmit  motion  with  a  constant 
velocity  ratio.  There  is,  however,  necessarily  a  slipping  of  the 
elements  of  the  surfaces  upon  each  other  parallel  to  themselves. 
Teeth  may  be  formed  on  these  pitch  surfaces,  and  they  may  be 
used  for  the  transmission  of  motion  between  shafts  that  are  not 
parallel  nor  in  the  same  plane.  Such  gears  are  called  "  Skew  Bevel 
Gears."  The  difficulties  of  construction  and  the  additional  friction 
due  to  slipping  along  the  elements,  make  them  undesirable  in  prac- 
tice, and  there  is  seldom  a  place  where  they  cannot  be  replaced  by 
some  other  form  of  connection. 

A  very  complete  discussion  of  the  subject  of  Skew  Bevel  Gears 
may  be  found  in  Prof.  McCord's  "  Kinematics." 

61.  Spiral  G-earing. —  If  line  contact  is  not  essential  there  is  much 
wider  range  of  choice  of  gears  to  connect  shafts  which  are  neither 
parallel  nor  intersecting.  A  and  B,  Fig.  63,  are  axes  of  rotation  in 
different  planes,  both  planes  being  parallel  to  the  paper.     Let  EF 


TOOTHED   WHEELS,    OR   GEARS.  65 

and  GH  be  cylinders  on  these  axes,  tangent  to  each  other  at  the 
point  S.  Any  line  may  now  be  drawn  through  S  either  between  A 
and  5,  or  coinciding  with  either  of  them.  This  line,  say  DS,  may 
be  taken  as  the  common  tangent  to  helical  or  screw  lines  drawn  on 
the  cylinders  EF  and  GH  \  or  helical  surfaces  may  be  formed  on 
both  cylinders,  DS  being  their  common  tangent  at  S.  Spiral  Gears 
are  thus  produced.  Each  one  is  a  portion  of  a  many-threaded 
screw.  The  contact  in  these  gears  is  point  contact ;  in  practice  the 
point  of  contact  becomes  a  very  limited  area. 

62.  When  the  angle  between  the  shafts  is  made  equal  to  90°, 
and  one  gear  has  only  one  or  two  threads,  it  becomes  a  special  case 
of  spiral  gearing  known  as  Worm  Gearing.  In  this  special  case  the 
gear  with  a  single  or  double  thread  is  called  the  worm,  while  the 
other  gear,  which  is  still  a  many-threaded  screw,  is  called  the  worm, 
wheel.  If  a  section  of  a  worm  and  worm  wheel  be  made  on  a  plane 
passing  through  the  axis  of  the  worm,  and  normal  to  the  axis  of 
the  worm  wheel,  the  form  of  the  teeth  will  be  the  same  as  that  of  a 
rack  and  pinion  ;  in  fact  the  worm,  if  mov^d  parallel  to  its  axis, 
would  transmit  rotary  motion  to  the  worm  wheel.  From  the  con- 
sideration of  racks  and  pinions  it  follows  that  if  the  involute  sys- 
tem is  used,  the  sides  of  the  worm  teeth  will  be  straight  lines. 
This  simplifies  the  cutting  of  the  worm,  because  a  tool  may  be  used 
capable  of  being  sharpened  without  special  methods.  If  the  worm 
wheel  were  only  a  thin  plate  the  teeth  would  be  formed  like  those 
of  a  spur  gear,  of  the  same  pitch  and  diameter.  But  since  the 
worm  wheel  must  have  greater  thickness,  and  since  all  other  sec- 
tions parallel  to  that  through  the  axis  of  the  worm,  as  CD  and  AB, 
Fig.  64,  show  a  different  form  and  location  of  tooth,  it  is  necessary 
to  make  the  teeth  of  the  worm  wheel  different  from  those  of  a  spur 
gear,  if  there  is  to  be  contact  between  the  worm  and  worm  wheel 
anywhere  except  in  the  plane  EF,  Fig.  64.  This  would  seem  to  in- 
volve great  difficulty,  but  it  is  accomplished  in  practice  as  follows  : 
A  duplicate  of  the  worm  is  made  of  tool  steel,  and  "  flutes"  are  cut 
in  it  parallel  to  the  axis,  thus  making  it  into  a  cutter,  which  is  tem- 
pered.    It  is  then  mounted  in  a  frame  in  the  same  relation  to  the 


bb  MACHINE    DESIGN. 

worm  wheel  that  the  worm  is  to  have  when  they  are  tinished  and 
in  position  for  working.  The  distance  between  centres,  however, 
is  somew^hat  greater,  and  is  capable  of  being  gradually  reduced. 
Both  are  then  rotated  with  the  required  velocity  ratio  by  means  of 
gearing  properly  arranged,  and  the  cutter  or  "hob"  is  fed  against 
the  worm  wheel  till  the  distance  between  centres  becomes  the 
required  value.  The  teeth  of  the  worm  wheel  are  "  roughed  out " 
before  they  are  "bobbed.*'  By  the  above  method  the  worm  is  made 
to  cut  its  own  worm  wheel.* 

Fig.  65  represents  the  half  section  of  a  worm.  If  it  is  a  single 
worm  the  thread  A,  in  going  once  around,  comes  to  B ;  twice 
around,  to  C  ;  and  so  on.  If  it  is  a  double  worm  the  thread  A,  in 
going  once  around,  comes  to  C,  while  there  is  an  intermediate 
thread,  B.  It  follows  that  if  the  single  worm  turns  through  one 
revolution  it  will  push  one  tooth  of  the  worm  wheel  with  which  it 
engages,  past  the  line  of  centres  ;  while  the  double  worm  will  push 
two  teeth  of  the  worm  wheel  past  the  line  of  centres.  The  single 
worm,  therefore,  musf  make  as  many  revolutions  as  there  are  teeth 
in  the  worm  wheel,  in  order  to  cause  one  revolution  of  the  worm 
wheel  ;  while  for  the  same  result  the  double  worm  only  needs  to 
make  half  as  many  revolutions.  The  ratio  of  the  angular  velocity 
of  a  single  worm  to  that  of  the  worm  wheel  with  which  it  engages 

is  =  -,  in  which  n  equals  the  number  of  teeth  in  the  worm  wheel. 
For  the  double  worm  this  ratio  is  -. 

Worm  gearing  is  particularly  well  adapted  for  use  where  it  is 
necessary  to  get  a  high  velocity  ratio  in  limited  space. 

The  pitch  of  a  worm  is  measured  parallel  to  the  axis  of  rotation. 
The  pitch  of  a  single  worm  is  p,  Fig.  65.  It  is  equal  to  the  circular 
pitch  of  the  worm  wheel.  The  pitch  of  a  double  worm  is  pi  =  2p=^ 
2  X  circular  pitch  of  the  worm  wheel. 

*  This  subject  is  fully  treated  in  Unwin's  "  Elements  of  Machine  Design," 
and  in  Brown  &  Sharpe's  *'  Treatise  on  Gearing." 


[Uiri7BRSIT7] 


TOOTHED   WHEELS,    OR   GEARS.  67 

63.  Design  of  Worm  Gears.  —  Let  £"  =  energy  to  be  transmitted 
through  the  worm  wheel  per  minute;  A^=  number  of  revolutions 
per  minute  ;  R  =  pitch  radius  of  the  worm  wheel ;  F  =^  force  at 
pitch  surface  of  the  worm  wheel.     Then 

E  =  2rtRN  X  F  =  space  factor  X  force  factor, 

jp  _     E     torsional  moment 

^^"^  ^~2^^RN~  R  • 

Hence,  if  E,  R,  and  N  are  given,  F  becomes  known,  and  an  approx- 
imate value  for  pitch  may  be  found  from  Table  I.     If  p  is  the 

diametral  pitch  thus  found,  the  corresponding  circular  pitch  =  - 

=  pitch  of  a  single  worm  to  mesh  with  the  worm   wheel.*     The 

pitch  of  a  double  worm  to  mesh  with  the  worm  wheel  =  — .     The 

P 
number  of  teeth,  n,  in  the  worm  wheel  =  2Rp.     For  a  single  worm 

71  71 

the  velocity  ratio  =  -  ;  for  double  worm  =  -.     The  rate  of  rotation 

Nn 
of  the  single  worm  ^  i\^n  ;  of  the  double  worm  = -^.     From  the 

energy,  Ei,  to  be  transmitted  per  minute  by  the  worm  shaft,  and  the 

rate  of  rotation,  N,,  the  moment  of  torsion  is  found  =7^-4^.    From 

2t:Ni 

this  a  suitable  belt  driving  mechanism  may  be  designed  by  methods 
to  be  given  later. 

64.  When  the  worm  and  worm  wheel  are  determined,  a  working 
drawing  may  be  made  as  follows  :  Draw  AB,  Fig.  66,  the  axis  of  the 
worm  wheel,  and  locate  0,  the  projection  of  the  axis  of  the  worm, 
and  P,  the  pitch  point.  With  0  as  a  centre  draw  the  pitch,  full 
depth,  and  addendum  circles,  G,  H,  and  K ;  also  the  arcs  CD  and 
EF,  bounding  the  tops  of  the  teeth  and  the  bottoms  of  the  spaces  of 
the  worm  wheel.  Make  the  angle  /5  =  90°.  Below  EF  lay  off  a 
proper  thickness  of  metal  to  support  the  teeth,  and  join  this  by  the 

*  This  value  must  be  made  such  that  it  may  be  cut  in  an  ordinary  lathe. 
See  next  page. 


68  MACHINE    DESIGN. 

web  LM  to  the  hub  N.  The  tooth  outlines  in  the  other  sectional 
view  are  drawn  exactly  as  for  an  involute  rack  and  pinion.  Full 
views  might  \  e  drawn,  but  they  involve  difficulties  of  construction, 
and  do  not  give  any  additional  information  to  the  workman. 

65.  Solution  from  Other  Data.  —  Two  shafts,  at  right  angles  to 
each  other  in  different  planes,  are  to  be  connected  by  means  of 
worm  gearing.  The  maximum  distance  between  them  is  fixed,  and 
its  value,  d,  is  given.  The  required  velocity  ratio,  r,  the  rate  of 
rotation  of  the  worm  shaft,  iV,  and  the  energy  to  be  transmitted  per 
minute,  E,  are  also  given.  It  is  required  to  design  the  gears.  A 
single  thread  worm  is  to  be  used.  Let  E  =  88000  ft. lbs.  per  min- 
ute ;  r  -^^  40,  i.  e.,  the  worm  makes  40  revolutions  per  revolution  of 
the  worm  wheel ;  d  =  8";  7\^  =  280  revolutions  per  minute.  The 
velocity  ratio  depends  upon  the  pitch  of  the  worm,  but  not  upon  its 
diameter.  Because,  whatever  the  diameter  of  the  worm,  it  j)ushes 
one  tooth  of  the  worm  wheel  past  the  line  of  centres  (Q7?,  Fig.  66) 
each  revolution.  The  pitch  diameter  of  the  worm  may  therefore  be 
any  convenient  value.  The  worm  wheel  must  have  40  teeth  in 
order  that  the  single  thread  worm  shall  turn  40  times  to  turn  it 
once.  The  number  of  threads  per  inch  of  the  worm,  measured 
axially,  is  the  reciprocal  of  the  pitch.  This  should  be  such  a  value 
that  it  may  be  cut  in  an  ordinary  lathe  without  special  arrange- 
ment of  the  change  gears.  Lathes  of  medium  size  are  capable  of 
cutting  1,  2,  8,  4,  etc.,  threads  per  inch.  The  circular  pitch  may 
therefore  be  1,  0"5,  0'384,  0*25,  etc.  In  the  case  considered  suppose 
that  the  pitch  diameter  of  the  worm  may  conveniently  be  about 
1"5".  The  corresponding  pitch  radius  of  the  worm  wheel  =  8" —  Vb" 
=  6"5".     Since  there  must  be  40  teeth,  it  follows  that  the  circular 

pitch,  =    Tn~  =  1'021.      This   should   be  exactly   1,   as  indicated 

above.  Let  circular  pitch  =  1,  and  check  for  strength.  The  tooth 
may  be  considered  as  a  cantilever,  as  in  the  case  of  spur  gear  teeth. 

Then  *S' =^  7-7^,  in  which  P=  force  at  the  pitch  surface,  I  =^  depth 

of  space,  6  =  distance  corresponding   to  the  arc  EF^  Fig,  66,  and 


:Uiri7BRSIT7] 


TOOTHED    WHEELS,    OR   GEARS.  69 

d  :=:  thickness  of  tooth  at  pitch  circle  ^=  ^  circular  pitch.  —  To  find 

P.     The   energy  =  38,000    ft. lbs.  per  minute  ;    N  =  rate  of    rota- 

280 
tion  of  the  worm  wheel  shaft  =  -jtt  =  ^  5    the  pitch  radius  of  the 

40 
worm  wheel   corrected   for   1   pitch  =  ^r-  =  6'37"  =  0*53    feet  ;   the 

torsional  moment  at  the  worm  wheel  shaft  =    ^   .^    =  Pr.     Hence, 

„       38000  33000  ,,,,^u        i      nav     ^      f^  ^^ 

^^2^^2O<-0^^7==^^^'^'"'    ^^'■'^'    ^^^■'- 

To  find  b.  The  pitch  radius  of  the  worm  corrected  for  1  pitch  = 
8  —  637  =  1'63.  The  radius  of  the  outside  of  the  worm  =  1'68  + 
addendum,  0"318,=  1*948  ;  the  arc  subtended  by  90°  on  this  radius 

fi  v  1414  V  064 
=  90°  X  00174  X  1-948  =  3-05 -- 6.     Hence  S  =      oa-       no-      = 

8"0o  X  0'2o 

7121  lbs.     If  cast  iron  were  to  be  used,  this  would  give  a  factor  of 

safety  =  _         =^  2'8.      This  is  too  small,   and   a    larger  circular 
/ 1^1 

pitch  would  need  to  be  used,  and  the  worm  would  have  to  be  cut  in 

a  lathe  capable  of  cutting  less   than  1   thread   per  inch.     If  steel 

casting  is  an  allowable  material  the  factor  of  safety  would  -=  ^ 

^=  7  +.     This  is  a  proper  value,  and  the  design  is  correct. 

This  method  of  design  applies  when  the  worm  wheel  is  cut  by 
a  "hob." 

66.  Compound  Spur  Gear  Chains.  —  Spur  gear  chains  may  be  com- 
pound, i.  e.,  they  may  contain  links  which  carry  more  than  two 
elements.  Thus  in  Fig.  67  the  links  a  and  d  each  carry  three 
elements.  In  the  latter  case  the  teeth  of  d  must  be  counted  as  two 
elements,  because  by  means  of  them  d  is  paired  with  both  h  and  c. 
In  the  case  of  the  three-link  spur  gear  chain  the  wheels  h  and  c 
meshed  with  each  other,  and  a  point  in  the  pitch  circle  of  c  moved 
with  the  same  linear  velocity  as  a  point  in  the  pitch  circle  of  b,  but 
in  the  opposite  direction.  In  Fig.  67  points  in  all  the  pitch  circles 
have  the  same  linear  velocity,  since  the  motion  is  equivalent   to 


70  MACHINE    DESIGN. 

rolling  together  of  the  pitch  circles  without  slipping  ;  hut  c  and  h 
now  rotate  in  the  same  direction.  Hence  it  is  seen  that  the  intro- 
duction of  the  wheel  d  has  reversed  the  direction  of  rotation,  with- 
out changing  the  velocity  ratio.  The  size  of  the  wheel  d,  which  is 
called  an  "  idler,"  has  no  effect  upon  the  motion  of  c  and  b.  It 
simply  receives,  upon  its  pitch  circle,  a  certain  linear  velocity  from 
c,  and  transmits  it  unchanged  to  b.  Hence  the  insertion  of  any 
number  of  idlers  does  not  affect  the  velocity  ratio  of  c  to  /),  but  each 
added  idler  reverses  the  direction  of  the  motion.  Thus,  with  an  odd 
number  of  idlers,  c  and  b  will  rotate  in  the  same  direction  ;  and 
with  an  even  number  of  idlers,  c  and  b  will  rotate  in  opposite  direc- 
tions. 

If  parallel  lines  be  drawn  through  the  centres  of  rotation  of  a 
pair  of  gears,  and  if  from  the  centres  distances  be  laid  off  on  these 
lines  inversely  proportional  to  the  angular  velocities  of  the  gears, 
then  a  line  joining  the  points  so  determined  will  cut  the  line  of 
centres  in  a  point  which  is  the  centro  of  the  gears.  In  Fig.  67, 
since  the  rotation  is  in  the  same  direction,  the  lines  have  to  be  laid 
off  on  the  same  side  of  the  line  of  centres.  The  pitch  radii  are 
inversely  proportional  to  the  angular  velocities  of  the  gears,  and 
hence  it  is  only  necessary  to  draw  a  tangent  to  the  pitch  circles  of  b 
and  c,  and  the  intersection  of  this  line  with  the  line  of  centres  is  the 
centro,  be,  of  c  and  b.  The  centroids  of  c  and  b  are  c,  and  b^,  circles 
through  the  point  be,  about  the  centres  of  e  and  b.  Obviously,  this 
four  link  mechanism  may  be  replaced  by  a  three  link  mechanism, 
in  which  c  is  an  annular  wheel  meshing  with  a  pinion  b.  The  four 
link  mechanism  is  more  compact,  however,  and  usually  more  con- 
venient in  practice. 

67.  The  other  principal  form  of  spur  gear  chain  is  shown  in 
Fig.  68.  The  wheel  d  has  two  sets  of  teeth  of  different  pitch  diam- 
eter, one  pairing  with  e,  and  the  other  with  b.  The  point  bd  now 
has  a  different  linear  velocity  from  cd,  greater  or  less  in  proportion 
to  the  ratio  of  the  radii  of  those  points.  The  angular  velocity  ratio 
may  be  obtained  as  follows  : 


r^  Of  THl        ^ 

[TIBlTBRSIT 


also 


TOOTHED   WHEELS,    OR   GEARS.  71 

angular  veloc.  d Ccd 

angular  veloc.  c       Dcd  ' 

angular  veloc.  h Dhd. 

angular  veloc.  d       Bhd  ' 


_^  -  .   .    .  angular  veloc.  h       Ccd  X  Dhd 

Multiplying,  angular  veloc.  c  =  /)c<JX56.r 

The  numerator  of  the  last  term  consists  of  the  product  of  the  radii 
of  the  "followers"  ;  and  the  denominator  consists  of  the  product  of 
the  radii  of  the  "  drivers."  The  diameters  or  numbers  of  teeth 
could  be  substituted  for  the  radii. 

In  general,  the  angular  velocity  of  the  first  driver  is  to  the 
angular  velocity  of  the  last  follower  as  the  product  of  the  number 
of  teeth  of  the  followers  is  to  the  product  of  the  number  of  teeth  of 
the  drivers.  This  applies  equally  well  to  compound  spur  gear 
trains  that  have  more  than  three  axes.  Therefore,  in  any  spur  gear 
chain  the  velocity  ratio  equals  the  product  of  the  number  of  teeth 
in  the  followers  divided  by  the  product  of  the  number  of  teeth  in 
the  drivers.  The  direction  of  rotation  is  reversed  if  the  number  of 
intermediate  axes  is  even,  and  is  not  reversed  if  the  number  is  odd. 
If  the  train  includes  annular  gears  their  axes  would  be  omitted 
from  the  number,  because  annular  gears  do  not  reverse  the  direc- 
tion of  rotation. 


CHAPTER    VI. 

CAMS. 

68.  A  machine  part  of  irregular  outline,  as  A,  Fig.  69,  may 
rotate  or  vibrate  about  an  axis  0,  and  communicate  motion  by  line 
contact  to  another  machine  part,  B.  A  is  called  a  cam.  A  cylin- 
der A,  Fig.  70,  having  a  groove  of  any  form  in  its  surface,  may 
rotate  about  its  axis,  CZ),  and  communicate  motion  to  another 
machine  part,  B.  ^  is  a  cam.  A  disc  A^  Fig.  71,  having  a  groove 
in  its  face,  may  rotate  about  its  axis,  0,  and  communicate  motion 
to  another  machine  part,  B.  ^  is  a  cam.  In  fact  it  is  only  a  mod- 
ification of  A,  Fig.  69.  In  designing  cams  it  is  customary  to  con- 
sider a  number  of  simultaneous  positions  of  the  driver  and  follower. 
The  cam  curve  can  usually  be  drawn  from  data  thus  obtained. 

69.  Case  I.  —  The  follower  is  guided  in  a  straight  line,  and  the 
contact  of  the  cam  with  the  follower  is  always  in  this  line.  The  line 
may  be  in  any  position  relatively  to  the  centre  of  .rotation  of  the 
cam  ;  hence  it  is  a  general  case.  The  point  of  the  follower  which 
bears  on  the  cam  is  constrained  to  move  in  the  line  MN,  Fig.  72. 
0  is  the  centre  of  rotation  of  the  cam.  About  0,  as  a  centre,  draw 
a  circle  tangent  to  MN  at  /.  Then  A,  B,  C,  etc.,  are  points  in  the 
cam.  When  the  point  ^  is  at  /  the  point  of  the  follower  which 
bears  on  the  cam  must  be  at  A'  ;  when  5  is  at  J  the  follower  point 
must  be  at  B';  and  so  on  through  an  entire  revolution.  Through 
A,  B,  C,  etc.,  draw  lines  tangent  to  the  circle.  With  0  as  a  centre, 
and  OA'  as  a  radius,  draw  a  circular  arc  A' A",  intersecting  the  tan- 
gent through  A  at  A".  Then  A"  will  be  a  point  in  the  cam  curve. 
For,  if  A  returns  to  J,  AA"  will  coincide  with  JA\  A"  will  coincide 
with  A\  and  the  cam  will  hold  the  follower  in  the  required  position. 


[ 


^A^   Of  THX         ^ 

[DBITJBSITT] 


CAMS.  73 

The  same  process  for  the  other  positions  locates  other  points  of  the 
cam  curve.  A  smooth  curve  drawn  through  these  points  is  tlie 
required  cam  outline.  Often,  to  reduce  friction,  a  roller  attached 
to  the  follower  rests  on  the  cam,  motion  being  communicated 
through  it.  The  curve  found  as  above  will  be  the  path  of  the  axis 
of  the  roller.  The  cam  outline  will  then  be  a  curve  drawn  inside 
of,  and  parallel  to,  the  path  of  the  axis  of  the  roller,  at  a  distance 
from  it  equal  to  the  roller's  radius.  Contact  between  the  follower 
and  the  cam  is  not  confined  to  the  line  MN  if  a  roller  is  used. 

70.  Case  II.  —  The  cam  engages  with  a  surface  of  the  follower, 
and  this  surface  is  guided  so  that  all  of  its  positions  are  parallel. 
The  method  given  is  due  to  Professor  J.  H.  Barr.  0,  Fig  78,  is  the 
centre  of  rotation  of  the  cam.  The  follower  surface  occupies  the 
successive  positions  1,  2,  3,  etc.,  when  the  lines  A,  B,  C,  etc.  of  the 
cam  coincide  with  the  vertical  line  through  C.  It  is  required  to 
draw  the  outline  of  a  cam  to  produce  the  motion  required.  Pro- 
duce the  vertical  line  through  0,  cutting  the  positions  of  the 
follower  surface  in  A\  B\  G',  etc.  With  0  as  a  centre  and  radii 
0B\  OC,  etc.,  draw  arcs  cutting  the  lines  B,  (7,  D,  etc.  in  the  points 
B'\  C'\  D'\  etc.  Position  1  is  the  lowest  position  of  the  follower 
surface  ;  therefore  A  must  be  in  contact  with  the  follower  surface  in 
the  vertical  line  through  0,  because  if  the  tangency  be  at  any  other 
point  the  motion  in  one  direction  or  the  other  will  lower  the  fol- 
lower, which  is  not  allowable.  A  is  therefore  one  point  in  the  cam 
curve.  Draw  a  line  MN  through  B"  at  right  angles  to  B"0,  and 
rotate  B"0  till  it  coincides  with  B'O.  Then  the  line  MN  will 
coincide  with  the  position  of  the  follower  surface  2J5'.  But  the  cam 
curve  must  be  tangent  to  this  line  when  B  coincides  with  B'O,  and 
therefore  the  line  MN  is  a  line  to  which  the  cam  curve  must  be  tan- 
gent. Similar  lines  may  be  drawn  through  the  points  C",  D'\  etc. 
Each  will  be  a  line  to  which  the  cam  curve  must  be  tangent.  There- 
fore, if  a  smooth  curve  be  drawn  tangent  to  all  these  lines,  it  will 
be  the  required  cam  outline. 

71.  Case  III.  —  This  is  the  same  as  Case  II,  except  that  the  posi- 
tions of  the  follower  surface  instead  of  being  parallel,  converge  to  a 
point,  0,  Fig,  74,  about  which  the  follower  vibrates.     The  solution 


74  MACHINE   DESIGN. 

is  the  same  as  in  Fig.  73,  except  that  the  angle  between  the  lines 
corresponding  to  MN^  Fig.  73,  and  the  radial  lines,  instead  of  being 
a  right  angle,  equals  the  angle  between  the  corresponding  position 
of  the  follower  surface  and  the  vertical. 

In  these  cases  the  cam  drives  the  follower  in  only  one  direction  ; 
the  force  of  gravity,  the  expansive  force  of  a  spring,  or  some  other 
force  must  hold  it  in  contact  with  the  cam.  To  drive  the  follower 
in  both  directions,  the  cam  surface  must  be  double,  i.  e.,  it  takes  the 
form  of  a  groove  engaging  with  a  pin  or  roller  attached  to  the  fol- 
lower, as  in  Fig.  71.  The  foregoing  principles  apply  to  the  laying 
out  of  the  curves. 

72.  Case  IV. —  To  lay  out  a  cam  groove  on  the  surface  of  a  cylin- 
der.—  A,  Fig.  75,  is  a  cylinder  which  is  to  rotate  continuously  about 
its  axis.  B  can  only  move  parallel  to  the  axis  of  A.  B  may  have  a 
projecting  roller  to  engage  with  a  groove  in  the  surface  of  A.  CD 
is  the  axis  of  the  roller  in  its  mid-position.  EF  is  the  development 
of  the  surface  of  the  cylinder.  During  the  first  quarter  revolution 
of  A,  CD  is  required  to  move  one  inch  toward  the  right  with  a 
constant  velocity.  Lay  off  (t^=  1",  and  H"/=  ^iTF,  locating /. 
Draw  GJ,  which  will  be  the  middle  line  of  the  cam  groove.  During 
the  next  half  revolution  of  A,  the  roller  is  required  to  move  two 
inches  toward  the  left  with  a  uniformly  accelerated  velocity.  Lay 
off  JX  =  2",  and  LM=  \KF.  Divide  LM  into  any  number  of 
equal  parts,  say  4.  Divide  JL  into  4  parts,  so  that  each  is  greater 
than  the  preceding  one  by  an  equal  increment.  This  may  be  done 
as  follows  :  1-1-2  +  3  +  4=  10.  Lay  off  from  /,  0"1  JL,  locating  a; 
then  0'2  JL  from  a,  locating  h  ;  and  so  on.  Through  a,  6,  and  c 
draw  vertical  lines  ;  through  m,  n,  and  o  draw  horizontal  lines. 
The  intersections  locate  d,  e,  and  /.  Through  these  points  draw 
the  curve  from  /to  Jf,  which  will  be  the  required  middle  line  of 
the  cam  groove.  During  the  remaining  quarter  revolution  the 
roller  is  required  to  return  to  its  starting  point  with  a  uniformly 
accelerated  velocity.  The  curve  MN  is  drawn  in  the  same  way  as 
JM.  On  each  side  of  the  line  GJMN  lay  off  parallel  lines,  their 
distance  apart  being  equal  to  the  diameter  of  the  roller.  Wrap  EF 
upon  the  cylinder,  and  the  required  cam  groove  is  located. 


H^"^  OF  thr"*^:^ 

;uiri7BiisrT 


^^^   Of  THB 

'UHIVBRSIT 


CHAPTER   VII. 

BELTS. 

73.  Transmission  of  Motion  by  Belts.  —  In  Fig.  76,  let  A  and  B  be 
two  cylindrical  surfaces,  free  to  rotate  about  their  axes  ;  let  CD  be 
their  common  tangent,  and  let  CD  represent  an  inextensible  con- 
nection between  the  two  cylinders.  Since  it  is  inextensible,  the 
points  D  and  C,  and  hence  the  surfaces  of  the  cylinders,  must  have 
the  same  linear  velocity.  Two  points  having  the  same  linear 
velocity,  and  different  radii,  have  angular  velocities  which  are  in- 
versely proportional  to  their  radii.  Hence,  since  the  surfaces  of 
the  cylinders  have  the  same  linear  velocity,  their  angular  velocities 
are  inversely  proportional  to  their  radii.  This  is  true  of  all  cylinders 
connected  by  inextensible  connectors.  Suppose  the  cylinders  to 
become  pulleys,  and  the  tangent  line  to  become  a  belt.  Let  CD' 
be  drawn  ;  this  becomes  a  part  of  the  belt,  making  it  endless,  and 
rotation  may  be  continuous.  The  belt  will  remain  always  tangent 
to  the  pulleys,  and  will  transmit  such  rotation  that  the  angular 
velocity  ratio  will  constantly  be  the  inverse  ratio  of  the  radii  of  the 
pulleys. 

The  case  considered  corresponds  to  a  crossed  belt,  but  the  same 
reasoning  applies  to  an  open  belt.  See  Fig.  77.  A  and  B  are  two 
pulleys,  and  CDD'C'C  is  an  open  belt.  Since  the  points  C  and  D 
are  connected  by  a  belt  that  is  practically  inextensible,  the  linear 
velocity  of  C  and  D  is  the  same  ;  therefore  the  angular  velocities  of 
the  pulleys  are  to  each  other  inversely  as  their  radii.  If  the  pulleys 
in  either  case  were  pitch  cylinders  of  gears  the  conditions  of  velocity 
would  be  the  same.  In  the  first  case,  however,  the  direction  of 
motion  is  reversed,  while  in  the  second  case  it  is  not.     Hence  the 


76  MACHINE    DESIGN. 

first  corresponds  to  gears  meshing  directly  with  each  other,  while 
the  second  corresponds  to  the  case  of  gears  connected  by  an  idler, 
or  to  the  case  of  an  annular  gear  and  pinion. 

Of  course  it  is  necessary  that  a  belt  should  have  some  thickness  ; 
and,  since  the  centre  of  pull  is  the  centre  of  the  belt,  it  is  necessary 
to  add  to  the  radius  of  the  pulley  half  of  the  thickness  of  the  belt. 
The  motion  communicated  by  means  of  belting,  however,  does  not 
need  to  be  absolutely  correct,  and  therefore  in  practice  it  is  usually 
customary  to  neglect  the  thickness  of  the  belt.  The  proportioning 
of  pulleys  for  the  transmission  of  any  required  velocity  ratio  is  now 
a  very  simple  matter. 

Illustration.  —  A  line  shaft  runs  150  revolutions  per  minute, 
and  is  supported  by  hangers  with  16"  "drop."  It  is  required  to 
transmit  motion  from  this  shaft  to  a  dynamo  to  run  1800  revolu- 
tions per  minute.  A  30"  pulley  is  the  largest  that  can  be  con- 
veniently used  with  16"  hangers.  Let  x  =  the  diameter  of  required 
pulley  for  the  dynamo  ;  then  from  what  has  preceded  z  -^-  30  ^= 
150  -r-  1800,  and  therefore  x  =  2'5".  But  a  pulley  less  than  4"  di- 
ameter should  not  be  used  on  a  dynamo.  Suppose  in  this  case  that 
it  is  6".  It  is  then  impossible  to  obtain  the  required  velocity  ratio 
with  one  change  of  speed,  i.  e.,  with  one  belt.  Two  changes  of 
speed  may  be  obtained  by  the  introduction  of  a  counter  shaft.  By 
this  means  the  velocity  ratio  is  divided  into  two  factors.  If  it  is 
wished  to  have  the  same  change  of  speed  from  the  line  shaft  to  the 
counter  as  from  the  counter  to  the  dynamo,  then  each  velocity  ratio 
would  be  V  (1800  -^  150)  =  ]  12  =  3*46.  But  this  gives  an  incon- 
venient fraction,  and  the  factors  do  not  need  to  be  equal.  Let  the 
factors  be  3  and  4.  See  Fig.  78.  A  represents  the  line  shaft,  B  the 
counter,  and  C  the  dynamo  shaft.  The  pulley  on  the  line  shaft  is 
30",  and  the  speed  is  to  be  three  times  as  great  at  the  counter,  and 
therefore  the  pulley  must  have  a  diameter  one-third  as  great,  =  10". 
The  pulley  on  the  dynamo  is  6"  diameter  and  the  counter  shaft  is 
to  run  one-fourth  as  fast,  and  therefore  the  pulley  on  the  counter 
opposite  the  dynamo  pulley  must  be  four  times  as  large  as  the 
dynamo  pulley,  =^  24". 


^4^   Of  THl 

lUHIVBESITT] 


BELTS.  77 

74.  A  belt  may  be  shifted  from  one  part  of  a  pulley  to  another 
by  means  of  pressure  against  the  side  which  advances  toward 
the  pulley.  Thus,  if  in  Fig.  79  the  rotation  be  as  indicated  by  the 
arrow",  and  side  pressure  be  applied  at  A,  the  belt  will  be  pushed  to 
one  side,  as  is  shown,  and  will  consequently  be  carried  into  some 
new  position  on  a  pulley  further  to  the  left  as  it  advances.  Hence, 
in  order  that  a  belt  may  maintain  its  position  on  a  pulley,  the  centre 
line  of  the  advancing  side  of  the  belt  must  be  perpendicular  to  the  axis 
of  rotation.  When  this  condition  is  fulfilled  the  belt  will  run  and 
transmit  the  required  motion,  regardless  of  the  relative  position  of 
the  shafts. 

75.  In  Fig.  80,  the  axes  AB  and  CD  are  parallel  to  each  other, 
the  above  stated  condition  is  fulfilled,  and  the  belt  will  run  cor- 
rectly ;  but  if  the  axis  CD  were  turned  into  some  new  position,  as 
CD',  the  side  of  the  belt  that  advances  toward  the  pulley  E,  cannot 
have  its  centre  line  in  a  plane  perpendicular  to  the  axis,  and  there- 
fore it  will  run  off.  But  if  a  plane  be  passed  through  the  line  CD, 
perpendicular  to  the  plane  of  the  paper,  then  the  axis  may  be  swung 
in  this  plane  in  such  a  way  that  the  necessary  condition  shall  be 
fulfilled,  and  the  belt  will  run  properly.  This  gives  what  is  known 
as  a  "twist"  belt,  and  when  the  angle  between  the  shafts  becomes 
90°,  it  is  a  "quarter  twist"  belt.  To  make  this  clearer,  see  Fig.  81. 
Rotation  is  transmitted  from  ^  to  5  by  an  open  belt,  and  it  is 
required  to  turn  the  axis  of  B  out  of  parallelism  with  that  of  A.  The 
direction  of  rotation  is  as  indicated  by  the  arrows.  Draw  the  line 
CD.  If  now  the  line  CD  be  supposed  to  pass  through  the  centre  of 
the  belt  at  C  and  Z),  it  may  become  an  axis,  and  the  pulley  B  and 
the  part  of  the  belt  FC  may  be  turned  about  it,  while  the  pulley  A 
and  the  part  of  the  belt  ED  remain  stationary.  During  this  motion 
the  centre  line  of  the  part  of  the  belt  CF,  which  is  the  part  that 
advances  toward  the  pulley  B  when  rotation  occurs,  is  always  in  a 
plane  perpendicular  to  the  axis  of  the  pulley  B.  The  part  ED, 
since  it  has  not  been  moved,  has  also  its  centre  line  in  a  plane  per- 
pendicular to  the  axis  of  A.  Therefore,  the  pulley  B  may  be  swung 
into  any  angular  position  about  CD  as  an  axis,  and  the  condition 
of  proper  belt  transmission  will  not  be  interfered  with. 


78  MACHINE    DESIGN. 

76.  If  the  axes  intersect,  the  motion  can  be  transmitted  between 
them  by  belting  only  by  the  use  of  "guide"  or  "idler"  pulleys. 
Let  AB  and  CD,  Fig.  82,  be  intersecting  axes,  and  let  it  be  required 
to  transmit  motion  from  one  to  the  other  by  means  of  a  belt  run- 
ning on  the  pulleys  E  and  F.  Draw  centre  lines  EK  and  FH 
through  the  pulleys.  Draw  the  circle,  G,  of  any  convenient  size, 
tangent  to  the  lines  EK  and  FH.  In  the  axis  of  the  circle  G,  let  a 
shaft  be  placed  on  which  are  two  pulleys,  their  diameters  being 
equal  to  that  of  the  circle  G.  These  will  serve  as  guide  pulleys  for 
the  upper  and  lower  sides  of  the  belt,  and  by  means  of  them  the 
centre  lines  of  the  advancing  parts  of  both  sides  of  the  belt  will  be 
kept  in  planes  perpendicular  to  the  axis  of  the  pulley  toward  which 
they  are  advancing,  the  belts  will  run  properly,  and  the  motion 
will  be  transmitted  as  required. 

77.  An  analogy  will  be  noticed  between  gearing  and  belting  for 
the  transmission  of  rotary  motion.  Spur  gearing  corresponds  to  an 
open  or  crossed  belt,  transmitting  motion  between  parallel  shafts. 
Bevel  gears  correspond  to  a  belt  running  on  guide  pulleys,  trans- 
mitting motion  between  intersecting  shafts.  Skew  bevel  and  spiral 
gears  correspond  to  a  "  twist "  belt,  transmitting  motion  between 
shafts  that  are  neither  parallel  nor  intersecting. 

78.  If  a  flat  belt  be  put  on  a  "  crowning  "  pulley,  as  in  Fig.  83, 
the  tension  on  AB  will  be  greater  than  on  CD,  the  belt  will  tend  to 
be  shifted  into  the  position  shown  by  the  dotted  lines  Eand  F,  and  as 
rotation  goes  on,  the  belt  will  be  carried  toward  the  high  part  of  the 
pulley,  i.  e.,  it  will  tend  to  run  in  the  middle  of  the  pulley.  This  is 
the  reason  why  nearly  all  belt  pulleys,  except  those  on  which  the 
belt  has  to  be  shifted  into  different  positons,  are  turned  "crown- 
ing." 

79.  Cone  Pulleys. —  In  performing  different  operations  on  a  ma- 
chine or  the  same  operations  on  materials  of  different  degrees  of 
hardness,  different  speeds  are  required.  The  simplest  way  of 
obtaining  them  is  by  the  use  of  cone  pulleys.  One  pulley  has  a 
series  of  steps,  and  the  opposing  pulley  has  a  corresponding  series 
of  steps.     By  shifting  the  belt  from  one  pair  to  another  the  velocity 


( 

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^4^    Of  TBOI^^ 

'UKI7BRSIT7J 


'-y^    Of  THIS         -^ 

[DHIVBRSIITl 


BELTS.  79 

ratio  is  changed.  Since  the  same  belt  is  used  on  all  the  pairs  of 
steps,  they  must  be  so  proportioned  that  the  belt  length  for  all  the 
pairs  shall  be  the  same  ;  otherwise  the  belt  would  be  too  tight  on 
some  of  the  steps  and  too  loose  on  others.  Let  the  case  of  a  crossed 
belt  be  first  considered.  The  length  of  a  crossed  belt  may  be 
expressed  by  the  following  formula  :  Let  L  =  length  of  the  belt ; 
d  =  distance  between  centres  of  rotation  ;  R  --=^  radius  of  the  larger 
pulley  ;  r  ^=  radius  of  the  smaller  pulley.  See  Fig.  84.  Then  L  = 
2^/d'—(R  -i-  ry^  (R^r)(7t  -j-2  arc  whose  sine  is  R -\- r  ^  d). 
In  the  case  of  a  crossed  belt,  if  the  size  of  steps  be  changed  so  that 
the  sum  of  their  radii  remains  constant,  the  belt  length  will  be  con- 
stant. For  in  the  formula  the  only  variables  are  R  and  r,  and  these 
terms  only  appear  in  the  formula  as  jR  -}-  r  ;  but  R  -\-  r  is  by 
hyothesis  constant.  Therefore  any  change  that  is  made  in  the 
variables  R  and  r,  so  long  as  their  sum  is  constant,  will  not  affect 
the  value  of  the  equation,  and  hence  the  belt  length  will  be  con- 
stant. It  will  now  be  easy  to  design  cone  pulleys  for  crossed  belt. 
Suppose  a  pair  of  steps  given  to  transmit  a  certain  velocity  ratio. 
It  is  required  to  find  a  pair  of  steps  that  will  transmit  some  other 
velocity  ratio,  the  length  of  belt  being  the  same  in  both  cases.  Let 
r  and  r'  =  radii  of  the  given  steps  ;  R  and  R'  =  radii  of  the  required 
steps  ;  r  -\-  r  =  R  -{-  R'  =  a  ;  the  velocity  ratio  of  R  to  R'  =  h. 
There  are  two  equations  between  R  and  R',  R  -f-  R'=  6,  and  R-\-  R' 
=  a.  Combining  and  solving,  it  is  found  that  J?'  =  aH-(l-fft), 
and  R  =  a  —  R'.  For  an  open  belt  the  formula  for  length  is  :  L  = 
2i/(/^  —  (R~ry  +  7t(R  +  r)  +  2  (R  —  r)  (arc  whose  sine  is  R  — 
r^d).  li  R  and  r  be  changed  as  before,  the  term  R  —  r  would  of 
course  not  be  constant,  and  two  of  the  terms  of  the  equation  would 
vary  in  value  ;  therefore  the  length  of  the  belt  would  vary.  The 
determination  of  cone  steps  for  open  belts  therefore  becomes  a  more 
difficult  matter,  and  approximate  methods  are  almost  invariably 
used. 

80.  The  following  graphical  approximate  method  is  due  to  Mr. 
C.  A.  Smith,  and  is  given,  with  full  discussion  of  the  subject,  in 
*' Transactions  of  the  American  Society  of  Mechanical  Engineers," 


80  MACHINE    DESIGN. 

Vol.  X,  p.  269.  Suppose  first  that  the  diameters  of  a  pair  of  cone 
steps  that  transmit  a  certain  velocity  ratio  are  given,  and  that  the 
diameters  of  another  pair  that  shall  serve  to  transmit  some  other 
velocity  ratio  are  required.  The  distance  between  centres  of  axes  is 
given.  See  Fig.  85.  Locate  the  pulley  centres  0  and  0\  at  the  given 
distance  apart  ;  about  these  centres  draw  circles  whose  diameters 
equal  the  diameters  of  the  given  pair  of  steps  ;  draw  a  straight  line 
(r/f,  tangent  to  these  circles  ;  at  /,  the  middle  point  of  the  line  of 
centres,  erect  a  perpendicular,  and  lay  off  a  distance  JK  equal  to 
the  distance  between  centres,  C,  multiplied  by  the  experimentally 
determined  constant  0814  ;  about  the  point  K  ^o  determined,  draw  a 
circular  arc  AB,  tangent  to  the  line  GH.  Any  line  drawn  tangent  to 
this  arc  will  be  the  common  tangent  to  a  pair  of  coneste})s  giving  the 
same  belt  length  as  that  of  the  given  pair.  For  example,  suppose 
that  OD  is  the  radius  of  one  step  of  the  required  pair  ;  about  0, 
with  a  radius  equal  to  OD,  draw  a  circle  ;  tangent  to  this  circle  and 
to  the  arc  AB,  draw  a  straight  line  DE  ;  about  0'  and  tangent  to 
DE,  draw  a  circle  ;  its  diameter  will  equal  that  of  the  required  step. 
But  suppose  that  instead  of  having  one  step  of  the  required  pair 
given,  to  find  the  other  corresponding  as  above,  a  pair  of  steps  are 
required  that  shall  transmit  a  certain  velocity  ratio,  =  r,  with  the 
same  length  of  belt  as  the  given  pair.  Suppose  OD  and  0' E  to  rep- 
resent the  unknown  steps.  The  given  velocity  ratio  equals  r.  But 
from    similar   triangles    CD  -f-  0' E  =  FO  ~  FO' .      Therefore   r  = 

FO  C  ^-  X 

TTTT,;    but  FO  =^  C  ^  X,  and  FO' =  x.      Therefore    r= ,  and 

FO  X. 

C 

Hence,  with    r   and  C  given,  the  distance  x  may  be 


found,  such  that  if  from  F  a  line  be  drawn  tangent  to  AB,  the  cone 
steps  drawn  tangent  to  it  will  give  the  velocit}^  ratio,  r,  and  a  belt 
length  equal  to  that  of  any  pair  of  cones  determined  by  a  tangent 
to  AB.  The  point  F  often  falls  at  an  inconvenient  distance.  The 
radii  of  the  required  steps  may  then  be  found  as  follows  :  Place  a 
straight-edge  tangent  to  the  arc  AB  and  measure  the  perpendicular 
distances  from  it  to  0  and  0'.  The  straight-edge  may  be  shifted 
until  these  distances  bear  the  required  relation  to  each  other. 


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BELTS.  81 

81.  Design  of  Belts. —  Fig.  86  represents  two  pulleys  connected 
by  a  belt.  When  no  moment  is  applied  tending  to  produce  rota- 
tion this  tension  in  the  two  sides  of  the  belt  is  equal.  Let  T.^  repre- 
sent this  tension.  If  now  an  increasing  moment,  represented  by  Rl, 
be  applied  to  the  driver,  its  effect  is  to  increase  the  tension  in  the 
lower  side  of  the  belt  and  to  decrease  the  tension  in  the  upper  side. 
With  the  increase  of  Rl  this  difference  of  tension  increases  till  it  is 
equal  to  P,  the  force  with  which  rotation  is  resisted  at  the  surface  of 
the  pulley.  Then  rotation  begins,*  and  continues  as  long  as  this 
equality  continues;  i.e.^a^  long  as  T^ — T.^=  P,  in  which  Ty=^ 
tension  in  the  driving  side,  and  T.^  =  tension  in  the  slack  side. 
The  tension  in  the  driving  side  is  increased  at  the  expense  of  that 

T  -{-  T. 

in  the  slack  side.      Therefore  — ^ — -  =  T... 

T 

To  find  the  value  of  ^.     The  increase  in  tension  from  the  slack 

side  to  the  driving  side  is  possible  because  of  the  frictional  resist- 
ance between  the  belt  and  pulley  surface.  Consider  any  element  of 
the  belt,  ds,  Fig.  88  {a).  It  is  in  equilibrium  under  the  action  of 
the  following  forces  :  T,  the  value  of  the  varying  tension  corres- 
ponding to  the  section,  acts  upon  one  end  of  ds  and  is  aided  by  dF. 
The  force  T-f  ^^^acts  upon  the  other  end.  From  the  action  of 
these  forces  there  results  a  normal  pressure  between  ds  and  pulley, 
=  pds,  in  which  p  =  the  pressure  per  linear  unit  of  belt.  Draw 
the  force  triangle,  {h)    Fig.  88.      It  is  an   isosceles   triangle,   and 

hence pds  =  (T -{- dT)0  ;    but  0--='^;  /.  pds  =-   ( ^  +  ^^ )^^  .    ^i/p 

vanishes  ;    .*.  p  =  -. 
r 

Since  the  force  triangle  is  an  isosceles  triangle,  it  follows  that 

T -^  dT=  T -}-  dF;  hence  dT  =  dF.     Suppose  that  rotation  occurs 

and  that  the  belt  slips  upon  the  pulley  at  a  rate  corresponding  to  a 

*  While  the  moving  parts  are  being  brought  up  1  o  speed  the  difference  of 
tension  must  equal  P-f-  force  necessary  to  produce  the  acceleration. 


82  MACHINE    DESIGN. 

T 

coefficient  of  friction  /.     Then  dF  -~fpds,  and  since  p  =  —, 

Tds 
.-.  dT^f^  hut  ds  =  rd(K 
r 

.-.   dT=fTdO; 


dO: 


T 

T 

-i  =  e/«,  where  e  =  the  Naperian  base. 

log  ^'  =  0-4343/«. 

-*  -2. 

«  is  in  7r  measure  and  equals  «  in  degrees  X  0'0174. 
82.  The  following  equations  are  established  : 

T,--T,  =  P  (1) 

T,-i^T,  =  2T,,  (2) 

^  =  e/«  or  log  I'  -=  0-4848  /.  ( 8 ) 

The  right  hand  members  of  (1)  and  (3)  can  usually  be  deter- 
mined ;  hence  the  value  of  Tj  (the  maximum  stress  in  the  belt) 
may  be  found,  and  proper  proportions  may  be  given  to  the  belt.  If 
W^  foot-pounds  per  minute  are  to  be  transmitted,  and  the  velocity 
of  the  rim  of  the  pulley  transmitting  this  power  in  feet  per  minute 
equal  S,  then  the  force  /*  equals  the  work  divided  by  the  velocity;  or, 

W 
P=^~.     The  value  of  «  is  found  from  the  diameters  of  the  pulleys 

and  their  distance  between  centres,  and  may  usuall}^  be  estimated 
accurately  enough.  The  value  of/,  the  coefficient  of  friction,  varies 
with  the  kind  of  belting,  the  material  and  character  of  surface  of 
the  pulley,  and  with  the  rate  of  slip  of  the  belt  on  the  pulle3\  ^^~ 
periments  made  at  the  laboratory  of  the  Massachusetts  Institute  of 


^-y^  Of  THl^^ 

IDSITBESITTl 


BELTS.  88 

Technology,  under  the  direction  of  Professor  Lanza,  indicate  that 
for  leather  belting  running  on  turned  cast  iron  pulleys,  the  rate  of 
slip  for  efficient  driving  is  about  three  to  four  feet  per  minute  ;  and 
also  that  the  coefficient  of  friction'  corresponding  to  this  rate  of  slip 
is  about  0-27.  The  value  08  may  be  used.  If  this  value  of  /  be 
used  the  slip  will  be  kept  within  the  above  limits  if  the  belt  be  put 

T  ^  T., 

on  with  a  proper  initial  tension,  =  T.^  =  -^-^ — '- ,  and  the  driving  of 

z 

the  belt  so  designed  will  be  satisfactory. 

83.  Problem.  —  A  single-acting  pump  has  a  plunger  8 "  =  0"666 
feet  in  diameter,  whose  stroke  has  a  constant  length  of  10'  =^  0*888 
feet.  The  number  of  strokes  per  minute  is  50.  The  plunger  is 
actuated  by  a  crank,  and  the  crank  shaft  is  connected  by  spur  gears 
to  a  pulley  shaft,  the  ratio  of  gears  being  such  that  the  pulley  shaft 
runs  800  revolutions  per  minute.  The  pulley  which  receives 
the  power  from  the  line  shaft  is  18"  in  diameter.  The  pressure  in 
the  delivery  pipe  is  100  lbs.  per  square  inch.  The  line  shaft  runs 
150  revolutions  per  minute,  and  its  axis  is  at  "a  distance  of  12 
feet  from  the  axis  of  the  pulley  shaft.  Since  the  line  shaft  runs 
half  as  fast  as  the  pulley  shaft,  the  diameter  of  the  pulley  on  the 
line  shaft  must  be  twice  as  great  as  that  on  the  pulley  shaft,  or  36". 
The  work  to  be  done  per  minute,  neglecting  the  friction  in  the  ma- 
chine, is  equal  to  the  number  of  pounds  of  water  pumped  per 
minute  multiplied  Ijy  the  head  in  feet  against  which  it  is  pumped. 
The  number  of  cubic  feet  of  water  per  minute  equals  the  displace- 
ment of  the  plunger  in  cubic  feet  multiplied  by  the  number  of 

strokes  per  minute  =  — — ^^ X  0'888  X  50  =  14*5,  and  therefore 

the  number  of  pounds  of  water  pumped  per  minute  =  14*5  X  62*4 
=  907.  One  foot  vertical  height  or  "head'"  of  water  corresponds 
to  a  pressure  of  0435  lbs.  per  square  inch,  and  therefore  100  lbs.  per 
square  inch  corresponds  to  a  "  head  "  of  100  -^  0'435  =  230  feet. 
The  work  done  per  minute  in  pumping  the  water  therefore  is  equal  to 
907  lbs.  X  230  feet  =  208,610  foot-pounds.  The  velocity  of  the  rim 
of  the  belt  pulley  =  300  X  1'5~  =  1410  feet  per  minute.     Therefore 


84  MACHINE   DESIGN. 

the  force  P^  T,~  T,  =  208,610  ft.lbs.  per  minute  --  1410  feet  per 
minute  =  147  lbs. 

To  tind  «,  see  Fig.  89.      Sin  0  =  ?^  =A-.=-  0-0625.      There- 
^  I  144 

fore  0  .=  8°  35';  «  =  180°  —  20  =  180°  —  7°  10'  ^  173°  nearly  ;   «  in 

r  measure  =  178  X  0*0174  =--  8-01. 

log  —'  =  0-4843  X  /  X  «  =  0-4843  X  0-3  X  3-01  =  0-3921. 

.-.  ^  =-  2-46  ;  P=T,—  T,  =  147. 

Combining  these  equations  T^  is  found  to  be  equal  to  246  lbs.,  = 
the  maximum  stress  in  the  belt.  E]xperiment  shows  that  70  lbs. 
per  inch  of  width  of  a  laced,  single  belt  is  a  safe  working  stress. 
Therefore  the  width  of  the  belt  =  246  -^  70  =  8-5".  The  friction  of 
the  machine  might  have  been  estimated  and  added  to  the  work  to 
be  done. 

84.  Problem.  —  A  sixty  horse-power  dynamo  is  to  run  1500  revo- 
lutions per  minute,  and  has  a  15"  pulley  on  its  shaft.  Power  is 
supplied  by  a  line  shaft  running  150  revolutions  per  minute.  A 
suitable  belt  connection  is  to  be  designed.  The  ratio  of  angular 
velocities  of  dynamo  shaft  to  line  shaft  is  10  to  1;  hence  the  diame- 
ter of  the  pulley  on  the  line  shaft  would  have  to  be  ten  times  as 
great  as  that  of  the  one  on  the  dynamo,  =  12-5  feet,  if  the  connec- 
tions were  direct.  This  is  inadmissible,  and  therefore  the  increase 
in  speed  must  be  obtained  by  means  of  an  intermediate,  or  counter 
shaft.  Suppose  that  the  diameter  of  the  largest  pulley  that  can  be 
used  on  the  counter  shaft  =  48".     Then  the  necessary  speed  of  the 

15 
counter  shaft  =  1500  X  j^  =  470  nearly.      The  ratio  of  diameters 

of  the  required  pulleys  for  connecting  the  line  shaft  and  the  counter 

470 
shaft  =  Y^^  =  8-13.     Suppose  that  a  60"  pulley  can  be  used  on  the 

line  shaft ;  then  the  diameter  of  the  required  pulley  for  the  counter 

shaft  will^^  — r^  =^  19"  nearly.     Consider  first  the  belt  to  connect 


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86  MACHINE    DESIGN. 

the  dynamo  to    the    counter   shaft.      The   work  =  60  X  33,000  = 
1,980,000  foot-pounds  per  minute;    the  rim  of  the  dynamo  pulley 
-1  n 

moves  '— ^  X  15C0  --  5890  feet  per  minute.      Therefore  1\  —  T.,= 

1^^  ^  .^^^  ^^^^      ,^,j^^  ^^.g  ^^  ^^^  counter  shaft  is  10  feet  from 
5890 

the  axis  of  the  dynamo,  and  as  before  sin  0  =  — - —  =^  — — — —  = 
0-1378.     Therefore  0  =  8°  nearly. 

a  =  180°  —  20=  164" 
a  in  rt  measure  =-  164  X  0-0174  =  2-85. 

log  T^  ^  0-4343  X  0-3  X  2-85  =  0*3713, 

^■  =  2-35. 

From  these  equations  T^  =  588  lbs.,  and  the  safe  width  of  the 
single  belting  =  583  --  70  ==  8-34";  say  8-5".  The  width  of  the  belt 
to  connect  the  line  shaft  to  the  counter  shaft  may  be  found  by  the 
same  method. 

85.  Table  II  is  given  to  save  the  above  calculations  for  each 
belt.  The  body  of  the  table  is  made  up  of  values  of  P,  the  driving 
force  at  the  pulley  surface.  To  use  the  table,  suppose  that  the 
smaller  angle  of  contact  of  the  belt  with  the  two  pulleys  considered 
is  known,  =  «°.  P  is  also  known.  Find  «",  or  the  nearest  smaller  value, 
in  the  horizontal  column  at  the  head  of  the  table.  In  the  vertical 
column  under  this  value  of  «,  find  P,  or  the  next  greater  value. 
Horizontally  opposite  this  in  the  first  vertical  column,  is  the  safe 
width  of  a  single  belt.  If  a  double  belt  is  to  be  used  the  value  found 
may  be  divided  by  2. 

86.  From  equation  (3),  p.  82,  it  follows  that  the  ratio  of  tensions, 

T 

-^,  when  the  belt  slips  at  a  certain  allowable  rate  (i.  e.,  when  /  is 

constant)  depends  only  upon  «.     This  ratio,  therefore,  is  indepen- 


BELTS.  a  / 

dent  of  the  initial  tension,  T^  ;  hence  "taking  up"  a  belt  does  not 

T 

change  -^.     The  difference  of  tension,  jT,  —  7*2  =  P,  is,  however,  de- 

pendent  on  T.^  Because  7),  the  normal  pressure  between  belt  and 
pulley,  varies  directly  as  T.^.  Then  since  dF  =fpds  =  dT,  it  follows 
that  (/r  varies  with  T^,  and  hence 


CdT=  T,—T,  =  P 


varies  with  T.^.  This  is  equivalent  to  saying  that  "taking  up"  a 
belt  increases  its  dri\ing  capacity,  and  "letting  it  out"  decreases 
its  driving  capacity. 

This  result  is  modified  because  another  variable  enters  the  prob- 
lem. If  T3  be  changed,  the  amount  of  slipping  changes,  and  the 
coefficient  of  friction  varies  directly  with  the  amount  of  slipping. 
Therefore,  an  increase  of  T.^  would  increase  j9  and  decrease/ in  the 
expression /j9fZs  =  dT,  and  the  converse  is  also  true.  This  is  prob- 
ably of  no  practical  importance. 

The  value  of  P  may  also  be  increased  by  increasing  either/,  the 
coefficient  of  friction,  or  «,  the  arc  of  contact ;    since  increase  of 

T 

either  increases  the  ratio  ~;  and  therefore  increases  T^ —  T.^  =:  P. 

Increasing  T.^  decreases  the  life  of  the  belt.  It  also  increases  the 
pressure  on  the  bearings  in  which  the  pulley  shaft  runs,  and  there- 
fore increases  frictional  resistance  ;  hence  a  greater  amount  of  the 
energy  supplied  is  converted  into  heat  and  lost  to  any  useful  pur- 
pose. Bat  if  T.^  be  kept  constant  and  /  or  «  be  increased,  the  driving 
power  is  increased  without  increase  of  pressure  in  the  bearings, 
because  this  pressure  =^27.^^=  constant.  AVhen  possible,  therefore, 
it  is  preferable  to  increase  P  by  increase  of /or  «,  rather  than  by 
increase  of  T. 

Application  of  belt  dressing  may  serve  sometimes  to  increase/. 

87.  If,  as  in  Fig.  86,  the  arrangement  is  such  that  the  upper 
side  of  the  belt  is  the  slack  side,  the  "sag"  of  the  belt  tends  to 

T 

increase  the  arc  of  contact,  and  therefore  to  increase  7;^.     If  the 


88  MACHINE   DESIGN. 

lower  side  is  the  slack  side,  the  belt  sags  away  from  the  pulleys 

T 
and  a  and  7=7  are  decreased. 

An  idler  pulley,  C,  may  be  used,  as  in  Fig.  90.  It  is  pressed  against 
the   belt  by  some  means.     Its  purpose  may  be  to  increase  P  by 

increasing  the  tension,  T.^,  =  -^— ^ — -.     In  this  case  friction  in  the 

bearings  is  increased.     Or  it  may  be  used  on  a  slack  belt  to  increase 

T 

the  angle  of  contact,  «,  the  ratio  -^,  and   therefore  P,  the  driving 

T  -\-  T. 

force.     In  this  case  the  value  of  T.^,  =    ^      — \  may    be   made   as 

small  a  value  as  is  consistent  with  driving,  and  hence  the  journal 
friction  may  be  small. 

Tighteners  are  sometimes  used  with  slack  belts  for  disengaging 
gear,  the  driving  pulley  being  vertically  below  the  follower. 

In  the  use  of  any  device  to  increase/  and  «,  it  should  be  remem- 
bered that  ^1  is  thereby  increased,  and  may  become  greater  than 
the  value  for  which  the  belt  was  designed.  This  may  result  in 
injury  to  the  belt. 

In  Fig.  91,  the  smaller  pulley,  A,  is  above  the  larger  one,  B.  A 
has  a  smaller  arc  of  contact,  and  hence  the  belt  would  slip  upon  it 
sooner  than  on  B.  The  weight  of  the  belt,  however,  tends  to  increase 
the  pressure  between  the  belt  and  A,  and  to  decrease  the  pressure 
between  the  belt  and  B.  The  driving  capacity  of  A  is  thereby  in- 
creased, while  that  of  B  is  diminished  ;  or,  in  other  words,  the 
weight  of  the  belt  tends  to  equalize  the  inequality  of  driving  power. 
If  the  larger  pulley  had  been  above,  there  would  have  been  a  ten- 
dency for  the  belt  weight  to  increase  the  inequality  of  driving 
capacity  of  the  pulleys.  The  conclusion  from  this,  as  to  arrange- 
ment of  pulleys,  is  obvious. 

88.  A  belt  resists  a  force  which  tends  to  bend  it.  Work  must  be 
done,  therefore,  in  bending  a  belt  around  a  pulley.  The  more  it  is 
bent  the  more  work  is  required.  Suppose  AB,  Fig.  92,  to  represent 
a  belt  which  moves  from  A  toward  B.     If  it  runs  upon  C  it  must 


[mtlTBRSIIT] 


_        BELTS.  89 

be  bent  more  than  if  it  runs  upon  D,  The  work  done  in  bending 
the  belt  is  converted  into  useless  heat  by  the  friction  between  the 
belt  fibres.  It  is  desirable,  therefore,  to  do  as  little  bending  as 
possible.  This  is  one  reason  why  large  pulleys  in  general  are  more 
efficient  than  small  ones.  The  resistance  to  bending  increases  with 
the  thickness  of  the  belt,  and  hence  double  belts  should  not  be  used 
on  small  pulleys  if  it  can  be  avoided. 

89.  Effect  of  Centrifugal  Force  of  Belts.  — In  Fig.  98,  as  the  belt 
reaches  a,  it  has  its  direction  of  motion  changed.  The  belt  tends  to 
move  on  in  a  straight  line,  and  therefore  resists  the  change  of  direc- 
tion. There  results  a  force  acting  radially  outward,  which  tends  to 
cause  the  belt  to  leave  the  pulley.     The  measure  of  this  force  per 

w       v^ 
linear  inch  of  belt=^r  =  -  X  -;  in  which  w  =  weight  of  belting  per 

linear  inch,  v  =  belt  velocity  in  feet  per  second,  g  =  82'2  feet  per 
second  acceleration,  r  =  pulley  radius.  As  the  velocity  of  the  belt 
is  increased,  w  and  r  remaining  constant,  c  will  increase,  and  will 
eventually  equal  the  radial  pressure  at  a  per  linear  inch  of  the  belt 
=  p.  With  further  increase  of  v  the  belt  would  leave  the  pulley  at 
a.  This  would  result  in  a  decrease  in  the  arc  of  contact,  and  hence 
a  decrease  in  the  driving  capacity  of  the  belt.  This  centrifugal 
force  is  the  same  for  every  linear  inch  of  the  belt  which  is  in  con- 
tact with  the  pulley,  i.  e.,  the  radial  force  acting  outward  is  constant 
from  a  around  to  b.  The  radial  pressure  79,  acting  inward,  however, 
increases  from  a  around  to  h  ;  hence  the  tendency  to  leave  the  pulley 
is  greatest  at  a.  Experience  shows  that  this  may  occur  in  practice, 
as  shown  in  Fig.  94,  the  angle  of  contact  being  reduced  from  «  to  /5. 
The  value  of  the  radial  pressure,  =  p,  due  to  belt  tension,  at  a,  the 
middle  of  the  first  inch  of  contact,  may  be  found.  The  value  is  less 
than  for  any  other  inch  of  contact,  because  it  increases  from  a 
around  to  h.  This  value  compared  with  the  centrifugal  force  found 
as  above  shows  the  tendency  for  the  belt  to  leave  the  pulley. 

To  find  p.  —  In  Fig.  95  the  first  inch  of  belt  in  contact  with  the 
pulley  at  a.  Fig.  93,  is  represented.  This  element  of  the  belt  sub- 
tends an  angle  f^,  whose  value  depends  on  the  radius  of  the  pulley. 


90  MACHINE    DESIGN. 

The  element  of  the  belt  is  in  equilibrium  under  the  influence  of 
three  forces,  T.^  +  ^T^,  T,^  -|-  ^F,  and  the  reaction  of  j?;  i.  e.,  the  pres- 
sure of  the  pulley  against  the  element  of  the  belt.  This  reaction  = 
j9,  and  its  line  of  action  (being  radial)  makes  equal  angles  with  the 
lines  of  action  of  T.^-^  ^ T.^  and  T.^  -j-  ^F.  The  angle  between  the 
lines  of  action  of  these  equal  forces  {T.^-j-  ^T.^  and  T.^^  ^F)  =  0. 
The  force  triangle  is  therefore  the  isosceles  triangle  shown.  In 
this  triangle 

7p-T-nr=  -- — q5  but  13  = ,  and  ^F=  -^ \ 

T.^-\-  -^F      sm  1^  2  s 

in  which  s  =  number  of  inches  of  contact  of  belt  with  pulley. 
Therefore 

T  —T 

T,-\~^^^—^'  sin  0 

P  = 


.     180° 
sm ~ 


Apply  this  to  the  problem,  on  page  84  :   T,  =  583  ;   T,  =  -^  =--  247. 

T,—T,  =  zm. 

s  =  aXr  =  164°  X  0-0174  X  7-5  =^  21-4". 

^=  angle  subtended  by  1"  on  7-5"  radius  =  360°  X  ^      „:^  =  7-64" 

=  7°  38'.     Sin  ^  =  0-1328. 

180°-^      172°  22'      __^, 
2 "~ 2 ^^^   ^^' 

sin  1?^;=^  =0-9977; 


(247  +  ^^)0-1328 


366- 

21-4/"""""      ^„ 0-01828 


^  = 0^9977 =  262-7  X  -^:^  =  34-7. 

To  find  c.  —  A  cubic  inch  of  belting  weighs  about  0*04  lb.  Single 
belting  is  about  0-25"  thick,  and  in  this  case  the  width  is  8-5".  The 
weight  of  belt  per  linear  inch  is  therefore  W=  0'2o  X  8*5  X  0-04= 
0-085  lbs. 


[nBIVBRSITT] 


BELTS.  91 

1500500 
12  X  60 

wv'      0-085  X  98-2-^ 


V  =     ,  /      ,.„     =  98*2  feet  per  second. 

40-8  lbs. 


^'  32-2  X^^ 

The  centrifugal  force  is  in  excess  at  (a)  the  middle  of  the  first  inch 
of  contact,  therefore,  by  an  amount  equal  to  40'8  lbs.  —  34*7  lbs.= 
6*1  lbs.  There  would  be  a  tendency  for  the  belt  to  lift.  This  is 
opposed,  however,  by  the  weight  of  the  belt.  If  the  slack  side  of 
the  belt  be  supposed  to  be  straight  and  horizontal,  one-half  its 
weight  will  be  supported  at  a.  The  distance  between  centres  of 
shafts  =  10  ft.  =^  120".  The  weight  W  of  the  slack  side  =  cubic  con- 
tents X  weight  per  cubic  unit,  =  0'25  X  8-5  X  120  X  0'04  =  10'2  lbs. 
=  W.  Half  of  this  aids  p  in  its  opposition  to  c  at  a.  Hence  the  out- 
ward radial  force  at  a  in  this  case  ■=  40*8  —  (34*7  +  5*1)  ==  1  lb.  If 
the  direction  of  rotation  were  reversed  the  slack  side  would  be  below 
and  the  outward  radial  force  at  b  would  equal  40*8  —  (34*7 — 5*1) 
=  11*2  lbs.  If  the  line  joining  the  centres  is  inclined,  as  in  Fig. 
97,  only  the  component  of  W  at  right  angles  to  the  belt,  =  P,  is 
effective  to  produce  inward  radial  pressure  at  a.  If  the  slack  side 
of  the  belt  becomes  vertical  P  becomes  =  0,  and  hence  the  weight 
has  no  effect. 

To  find  the  velocity  of  belt  at  which,  under  given  conditions, 
the  belt  just  tends  to  leave  the  pulley.  —  Let  w  =  radial  force  at  a 
due  to  belt  weight.  The  belt  just  tends  to  leave  the  pulley  when 
the  sum  of  the  inward  radial  forces  =  the  sum  of  the  outward  radial 

forces  ;  or  when  c  =p  ±  w;  substituting  value  of  c  = ,  in  which 

gr 

W=  weight   of   one  lineal   inch  of   the  belt,  and   solving  for  v  =^ 

(f)  zt  ly )  QT 

^^ —      '  ^  .     If  a  belt  tends   to  leave  the  pulley,  running  under 

given  conditions,  it  would  seem  that  increasing  the  radius  of  the 
pulley  upon  which  the  slack  side  runs  would  reduce  the  centrifugal 

force,  since  c  oc  - ;  but  in  order  to  keep  the  shaft  running  at  the  same 


4 


92  MACHINE    DESIGN. 

rate  as  before,  the  belt  must  run  faster  and  c  oc  v'\  Also,  since  the 
moment  to  produce  rotation  is  constant,  the  force  (with  the  increase 
of  lever  arm  due  to  increased  size  of  pulley)  is  less,  and  hence 
(unless  a  wider  belt  than  is  necessary  is  used)  the  width  of  the  belt 
and  hence  its  weight  per  linear  inch  must  be  reduced,  and  c  oc  iv. 

In  the  design  of  belting  care  should  be  taken  not  to  make  the 
distance  between  the  shafts  carrying  the  pulleys  too  small,  especially 
if  there  is  the  possibility  of  sudden  changes  of  load.  Belts  have 
some  elasticity,  and  the  total  yielding  under  any  given  stress  is 
proportional  to  the  length,  the  area  of  cross-section  being  the  same. 
Therefore  a  long  belt  becomes  a  yielding  part,  or  spring,  and  its 
yielding  may  reduce  the  stress  due  to  a  suddenly  applied  load  to  a 
safe  value  ;  whereas  in  the  case  of  a  short  belt,  with  other  conditions 
exactly  the  same,  the  stress  due  to  much  less  yielding  might  be 
sufficient  to  rupture  or  weaken  the  joint. 


CHAPTER  VIII. 

.     DESIGN    OF   FLY-WHEELS. 

90.  Often  in  machines  there  is  capacity  for  uniform  effort,  but 
the  resistance  fluctuates.  In  other  cases  a  fluctuating  effort  is 
applied  to  overcome  a  uniform  resistance,  and  yet  in  both  cases  a 
more  or  less  uniform  rate  of  motion  must  be  maintained.  When 
this  occurs,  as  has  been  explained,  a  moving  body  of  considerable 
weight  is  interposed  between  effort  and  resistance,  which,  because  of 
its  weight,  absorbs  and  stores  up  energy  with  increase  of  velocity 
when  the  effort  is  in  excess,  and  gives  it  out  with  decrease  of 
velocity  when  the  resistance  is  in  excess.  This  moving  body  is 
usually  a  rotating  body,  called  a  fly-wheel. 

To  fulfill  its  office  a  fly-wheel  must  have  a  variation  of  velocity  ; 
because  it  is  by  reason  of  this  variation  that  it  is  able  to  store  and 
give  out  energy.  The  kinetic  energy,  E,  of  a  body  whose  weight  is 
W,  moving  with  a  velocity  v,  is  expressed  by  the  equation 

^=%- 

To  change  E,  with  TF  constant,  v  must  vary.  The  allowable  varia- 
tion of  velocity  depends  upon  the  work  to  be  accomplished.  Thus, 
the  variation  in  an  engine  running  electric  lights,  or  spinning 
machinery,  should  be  very  small ;  probably  not  greater  than  a  half 
of  one  per  cent.  While  a  pump  or  a  punching  machine  may  have 
a  much  greater  variation  without  interfering  with  the  desired  re- 
sult. If  the  maximum  velocity,  i\,  of  the  fly-wheel  rim,  and  the 
allowable  variation  are  known,  the  minimum  velocity,  v.^,  becomes 
known  ;  and  the  energy  that  can  be  stored  and  given  out  with  the 


94  MACHINE    DESIGN. 

allowable  change  of  velocity  is  equal  to  the  difference  of  kinetic 
energy  at  the  two  velocities. 

The  general  method  for  fiy-wheel  design  is  as  follows  :  Find  the 
maximum  energy  due  to  excess  or  deficiency  of  effort  during  a  cycle 
of  action,  =  £".  Assume  a  convenient  mean  diameter  of  fly-wheel 
rim.  From  this  and  the  given  maximum  rotative  speed  of  the  fly- 
wheel shaft,  find  v^.  From  v^  and  the  given  allowable  variation  of 
velocity,  find  v.^.     Solve  the  above  equation  for  W,  thus  : 

v^^  —  v./ 

Substitute  the  values  of  E,  -yj,  v.^,  and  g  --  32"2  ft.  per  second.  Whence 
ir  becomes  known,  =  weight  of  fly-wheel  rim.  The  weight  of  rim 
only  will  be  considered  ;  the  other  parts  of  the  wheel  being  nearer 
the  axis  have  less  velocity,  and  less  capacity  per  pound  for  storing 
energy.  Their  effect  is  to  reduce  slightly  the  allowable  variation 
of  velocity. 

91.  Problem.  —  In  a  punching  machine  the  belt  is  capable  of 
applying  a  uniform  torsional  effort  to  the  shaft ;  but  most  of  the 
time  it  is  only  required  to  drive  the  moving  parts  of  the  machine 
against  frictional  resistance.  At  intervals,  however,  the  punch 
must  be  forced  through  metal  which  offers  shearing  resistance  to  its 
action.  Either  the  belt  or  fly-wheel,  or  the  two  combined,  must  be 
capable  of  overcoming  this  resistance.  A  punch  makes  30  strokes 
per  minute,  and  enters  the  die  ^".  It  is  required  to  punch  |"  holes 
in  steel  plates  i"  thick.  The  shearing  strength  of  the  steel  is  about 
50,000  pounds  per  square  inch.  When  the  punch  just  touches  the 
plate  the  surface  which  offers  shearing  resistance  to  its  action 
equals  the  surface  of  the  hole  which  results  from  the  punching,  =^ 
Tcdty  in  which  d  =  diameter  of  hole  or  punch,  t  =  thickness  of  plate. 
The  maximum  shearing  resistance,  therefore,  equals  7:|  X  ^  X  50000 
=  58800  lbs.  As  the  punch  advances  through  the  plate  the  resist- 
ance decreases,  because  the  surface  in  shear  decreases,  and  when  the 
punch    just   passes   through   the  resistance  becomes  zero.     If   the 


DESIGN    OF    FLY-WHEELS.  95 

change  of  resistance  be  assumed  uniform  (which  would  probably  be 

approximately  true)  the  mean  resistance  to  punching  would  equal 

,,  .  .  ,  ,  .  .  .  ,  ^  58800  +  0 
the  maximum   resistance  +  minimum  resistance,  -^  z,  = 

=  29400.  The  radius  of  the  crank  which  actuates  the  punch  =  2". 
In  Fig.  98  the  circle  represents  the  path  of  the  crank-pin  centre. 
Its  vertical  diameter  then  represents  the  travel  of  the  punch.  If 
the  actuating  mechanism  be  a  slotted  cross-head,  as  is  usual,  it  is 
a  case  of  harmonic  motion,  and  it  may  be  assumed  that  while  the 
punch  travels  vertically  from  A  to  B,  the  crank-pin  centre  travels 
in  the  semicircle  ACB.  Let  BD  and  DE  each  =:  ^  inch.  Then 
when  the  punch  reaches  E  it  just  touches  the  plate  to  be  punched, 
which  is  Y  thick,  and  when  it  reaches  D  it  has  just  passed  through 
the  plate.  Draw  the  horizontal  lines  EF  and  DG  and  the  radial 
lines  OG  and  OF.  Then,  while  the  punch  passes  through  the  plate, 
the  crank-pin  centre  moves  from  F  to  G,  or  through  an  angle  (in 
this  case)  of  19°.  Therefore  the  crank  shaft  ^,  Fig.  99,  and  attached 
gear  rotate  through  19°  during  the  action  of  the  punch.  The  ratio 
of  angular  velocity  of  the  pinion  and  the  gear  =  the  inverse  ratio  of 

An 
pitch  diameters  =  t-^  =  5.     Hence  the  shaft  B  rotates  through  an 

angle  =  19°  X  5  =  95°  during  the  action  of  the  punch.  If  there 
were  no  fly-wheel  the  belt  would  need  to  be  designed  to  overcome 
the  maximum  resistance ;  i.  e.,  the  resistance  at  the  instant  when 
the  punch  is  just  beginning  to  act.  This  would  give  for  this  case  a 
double  belt  about  20"  wide.  The  need  for  a  fly-wheel  is  therefore 
apparent.  Assume  that  the  fly-wheel  may  be  conveniently  36" 
mean  diameter,  and  that  a  single  belt  5"  wide  is  to  be  used.  The  allow- 
able maximum  tension  is  then-— 5  X  allowable  tension  per  inch  of 
width   of  single  belting  =r=  5  X  70  =  350  lbs.=  r,.     Then  from  the 

equation  ^  =  e/%iia=\SO%  ^' =2-56;  hence  7^,  =  ^^  -  |g^  = 

1365,  and  T^—  T^  =  213'5  lbs.  =  the  driving  force  at  the  surface 
of  the  pulley.  Assume  that  the  frictional  resistance  of  the  machine 
is  equivalent  to  25  lbs.  applied  at  the  pulley  rim.  Then  the  belt 
can  exert  213'o  lbs.  —  25  ^  188*5  lbs.,  :=  P,  to  accelerate  the  fly-wheel 


96  MACHINE   DESIGN. 

or  to  do  the  work  of  punching.  Assume  variation  of  velocity:^  10 
per  cent.  The  work  of  punching  =  the  mean  resistance  offered  to 
the  punch  multiplied  by  the  space  through  which  the  punch  acts,  = 

5?|22  X  0-5  =  14700  inch-pounds  =  1220  foot-pounds.     The  pulley 

shaft  moves  during  the  punching  through  95°,  and  the  driving  ten- 
sion  of   the   belt,  =  P=  188*5   lbs.,  does  work  =/*  X  space  moved 

95° 
through  during  the  punching  =  188'5  lbs.  X  '^d  ^—  =  188'5  lbs.  X  ^ 

95 
X  2  ft.  X  i^7^  =  311  foot-pounds.     The  work  left  for  the  fly-wheel 

to  give  out  with  a  reduction  of  velocity  of  10  per  cent.  =  1220  —  311 
=  909  foot-pounds.  Let  v^  =  maximum  velocity  of  fly-wheel  rim  ; 
v.^  =  minimum  velocity  of  fly-wheel  rim  ;  W=  weight  of  the  fly- 
wheel rim.    The  energy  it  is  capable  of  giving  out,  while  its  velocity 

is  reduced  from  v,  to  v.,,  =         ^^ '—,  and  the  value  of  W  must  be 

such  that  this  energy  given  out  shall  equal  909  foot-pounds.  Hence 
the  following  equation  may  be  written  : 

^^-^ — ^^  =  909. 

Therefore  W=  ^M^l^l. 

The  punch  shaft  makes  30  revolutions  per  minute  and  the  pulley 

shaft  30  X  5  ^  150  ^^  N  revolutions  per  minute.     Hence  v^  in  feet 

Nd- 
per  second  =  —^  ;  d  being  fly-wheel  diameter  in  feet, 

150  X  3:r 

^^^^60-  ^^^^• 

^;.^  ==  0-90  i;i=:  21-1. 

v^^  =  552  ;     v./  =  446  ;     v;'  —  v.J'  =  106. 

^  ,„      909  X  2  X  32-2       „_  ,, 

Hence  W= -7- =  551  lbs. 

lUo 


[UiriTBIlSITT] 


DESIGN    OF    FLY-WHEELS.  97 

To  proportion  the  rim.— A  cubic  inch  of  cast  iron  weighs  026  lbs.; 

551 
hence  there  must  be  ^-  =  2120  cu.  in.     The  cubic  contents  of  the 

rim  =  mean  diameter  X  ^r  X  its  cross-sectional  area>  A,  =  2120  cu. 

,                             .          2120         ^„_ 
in.  ;  hence  A  -—  ^r^ =  18  8    sq.  m. 

If  the  cross-section  were  made  square  its  side  would  =  i/18*8  ==- 
4-34". 

92.  Pump  Fly-Wheel. — The  belt  for  the  pump,  p.  83,  is  designed 
for  the  average  work.  A  fly-wheel  is  necessary  to  adapt  the  vary- 
ing resistance  to  the  capacity  of  the  belt.  The  rate  of  doing  work 
on  the  return  stroke  (supposing  no  resistance  due  to  suction)  is 
only  equal  to  the  frictional  resistance  of  the  machine!  During  the 
working  stroke  the  rate  of  doing  work  varies  because  the  velocity  of 
the  plunger  varies,  although  the  pressure  is  constant.  The  rate  of 
doing  work  is  a  maximum  when  the  velocity  of  the  plunger  is 
greatest.  In  Fig.  100,^  is  the  velocity  diagram;  B  is  the  force 
diagram  ;  C  is  the  tangential  diagram  drawn  as  indicated  on  pages 
35-36.  The  belt,  3'5"  wide,  is  capable  of  applying  a  tangential 
force  of  147  lbs.  to  the  18"  pulley  rim.  The  velocity  of  the  pulley 
rim  ^  -  1*5  X  300  =  1410.  The  velocity  of  the  crank-pin  axis  = 
TT  0-833  X  50  ^  130-8.     Therefore  the  force  of  147  lbs.  at  the  pulley 

rim  corresponds  to  a  force  =  147  X  ^^jk^  =  1585  lbs.  applied  tan- 

gentially  at  the  crank-pin  axis.  This  may  be  plotted  as  an 
ordinate  upon  the  tangential  diagram  C,  from  the  base  line  A^A'^, 
using  the  same  force  scale.  Through  the  upper  extremity  of  this 
ordinate  draw  the  horizontal  line  DE.  The  area  between  DE  and 
A^A'i  represents  the  work  the  belt  is  capable  of  doing  during  the 
working  stroke.  During  the  return  stroke  it  is  capable  of  doing  the 
same  amount  of  work.  But  this  work  must  now  be  absorbed  in 
accelerating  the  fly-wheel.  Suppose  the  plunger  to  be  moving  in 
the  direction  shown  by  the  arrow.  From  E  to  F  the  effort  is  in 
excess  and  the  fly-wheel  is  storing  energy.     From  F  to  G  the  resist- 


98  MACHINE    DESIGN. 

ance  is  in  excess  and  the  fly-wheel  is  giving  out  energy.  The  work 
the  fly-wheel  must  be  capable  of  giving  out  with  the  allowable 
reduction  of  velocity  is  that  represented  by  the  area  under  the  curve 
above  the  line  FG.  From  G  to  D,  and  during  the  entire  return 
stroke,  the  belt  is  doing  work  to  accelerate  the  fly-wheel.  This  work 
becomes  stored  kinetic  energy  in  the  fly-wheel.  Obviously  the 
following  equation  of  areas  may  be  written  : 

X,EF  +  XGD  4-  XHKX,  =-  GMF. 

The  left  hand  member  of  this  equation  represents  the  work  done  by 
the  belt  in  accelerating  the  fly-wheel ;  the  right  hand  member 
represents  the  work  given  out  by  the  fly-wheel  to  help  the  belt. 

The  work  in  foot-pounds  represented  by  the  area  GMF  may  be 
equated  with  the  difference  of  kinetic  energy  of  the  fly-wheel  at 
maximum  and  minimum  velocities.  To  find  the  value  of  this 
work:  One  inch  of  ordinate  on  the  force  diagram  represents  4260 
lbs. ;  one  inch  of  abscissa  represents  0*2245  ft.  Therefore  one 
square  inch  of  area  represents  4260  lbs.  X  02245  ==  956'37  foot- 
pounds. The  area  GMF  =^1'^  sq.  in.  Therefore  the  work  = 
956-37  X  1-6  =r=  1530  foot-pounds  =  E.       The  difference  of  kinetic 

W 
energy  =  — {v^  —  v./)  =:  1530  ;   TT  equals   the   weight    of   the   fly- 
wheel rim.     Hence 

1530  X  32-2 


W=^ 


V.:' 


Assume  the  mean  fly-wheel  diameter  =  2*5  ft.  It  will  be  keyed  to 
the  pulley  shaft,  and  will  run  300  revolutions  per  minute,  ==  5 
revolutions  per  second.  The  maximum  velocity  of  fly-wheel  rim  =- 
2'57r  X  5  =  39*15  =  t'l.  Assume  an  allowable  variation  of  velocity, 
=--  5  per  cent.  Then  v.,  =  39*15  X  0*95  =  37*19  ;  v;'  =  1532*7  ; 
v./  =  1383*1 ;  V,'  —  v./  =  149*6.     Hence 

^      1530  X  32*2       _Q  ,, 
^^        149*6        -^^^^^^- 


i    /yf.^^. 


Z<^  at  THi     ^ 
[tJHITBESIITl 


DESIGN    OF    FLY-WHEELS.  99 

There  must  be  329  ~  0-26  cubic  inches  in  the  rim  =:=  1262.  The 
pitch  circumference  =^  30  X  ~  =  94*2".  Hence  the  cross-sectional 
area  of  rim  =  1262  -^  94*2  =  13'4  -j-.  The  rim  may  be  made 
3"  X  4-5". 

The  frictional  resistance  of  the  machine  is  neglected.  It  might 
have  been  estimated  and  introduced  into  the  problem  as  a  constant 
resistance. 

93.  Steam  Engine  Fly-Wheel.  —  From  given  data  draw  the  indi- 
(  ator  card  as  modified  by  the  acceleration  of  reciprocating  parts, 
^^ee  page  35  and  Fig.  30.  From  this,  and  the  velocity  diagram, 
construct  the  diagram  of  tangential  driving  force,  Fig.  31.  Measure 
the  area  of  this  diagram  and  draw  the  equivalent  rectangle  on  the 
same  base.  This  rectangle  represents  the  energy  of  the  uniform 
resistance  during  one  stroke  ;  while  the  tangential  diagram  repre- 
sents the  work  done  by  the  steam  upon  the  crank-pin.  The  area 
of  the  tangential  diagram  which  extends  above  the  rectangle  repre- 
sents the  work  to  be  absorbed  by  the  fly-wheel  with  the  allowable 
variation  of  velocity.  Find  the  value  of  this  in  foot-pounds,  and 
equate  it  to  the  expression  for  difference  of  kinetic  energy  at  maxi- 
mum and  minimum  velocity.     Solve  for  TF,  the  weight  of  fly-wheel. 


CHAPTER   IX. 

RIVETED    JOINTS. 

94.  A  rivet  is  a  fastening  used  to  unite  metal  plates  or  rolled 
structural  forms,  as  in  boilers,  tanks,  built-up  machine  frames,  etc. 
It  consists  of  a  head,  A,  Fig.  101,  and  a  straight  shank,  B.  It  is  in- 
serted, usually  red-hot,  into  holes,  either  drilled  or  punched  in  the 
parts  to  be  connected,  and  the  projecting  end  of  the  shank  is  then 
formed  into  a  head  (see  dotted  lines)  either  by  hand  or  machine 
riveting.  A  rivet  is  a  permanent  fastening  and  can  only  be  removed 
by  cutting  off  the  head.  A  row  of  rivets  joining  two  members  is 
called  a  riveted  joint  or  seam  of  rivets.  In  hand  riveting  the  project- 
ing end  of  the  shank  is  struck  a  quick  succession  of  blows  with 
hand  hammers  and  formed  into  a  head  by  the  workman.  A  helper 
holds  a  sledge  or  "dolly  bar"  against  the  head  of  the  rivet.  In 
"button  set"  or  "snap"  riveting,  the  rivet  is  struck  a  few  heavy 
blows  with  a  sledge  to  "upset"  it.  Then  a  die  or  "button  set," 
Fig.  102,  is  held  with  the  spherical  depression,  B,  upon  the  rivet ; 
the  head  A  is  struck  with  the  sledge,  and  the  rivet  head  is  thus 
formed.  In  machine  riveting  a  die  similar  to  B  is  held  firmly  in 
the  machine  and  a  similar  die  opposite  to  it  is  attached  to  the  pis- 
ton of  a  steam,  hydraulic,  or  pneumatic  cylinder.  A  rivet,  properly 
placed  in  holes  in  the  members  to  be  connected,  is  put  between  the 
dies  and  pressure  is  applied  to  the  piston.  The  movable  die  is 
forced  forward  and  a  head  formed  on  the  rivet. 

The  relative  merits  of  machine  and  hand  riveting  have  been 
much  discussed.  Either  method  carefully  carried  out  will  produce 
a  good  serviceable  joint.  If  in  hand  riveting  the  first  few  blows  be 
light  the  rivet  will  not  be  properly  upset,  the  shank  will  be  loose  in 


RIVETED   JOINTS.  lOl 

the  hole,  and  a  leaky  rivet  results.  If  in  machine  riveting  the  axis 
of  the  rivet  does  not  coincide  with  the  axis  of  the  dies,  an  off-set 
head  results.  See  Fig.  103.  In  large  shops  where  w^ork  must  be 
turned  out  economically  in  large  quantities,  machines  must  be  used. 
But  there  are  always  places  inaccessible  to  machines,  where  the 
rivets  must  be  driven  by  hand.  Holes  for  the  reception  of  rivets 
are  usually  punched,  although  for  thick  plates  and  very  careful 
work  they  may  be  sometimes  drilled.  If  a  row  of  holes  be  punched 
in  a  plate,  and  a  similar  row  as  to  size  and  spacing  be  drilled  in  the 
same  plate,  testing  to  rupture  will  show  that  the  punched  plate  is 
weaker  than  the  drilled  one.  If  the  punched  plate  had  been 
annealed  it  would  have  been  nearly  restored  to  the  strength  of  the 
drilled  one.  If  the  holes  had  been  punched  ^"  small  in  diameter 
and  reamed  to  size,  the  plate  would  have  been  as  strong  as  the 
drilled  one.  These  facts,  which  have  been  experimentally  deter- 
mined, point  to  the  following  conclusions  :  First,  punching  injures 
the  material  and  produces  weakness.  Second,  the  injury  is  due  to 
stresses  caused  by  the  severe  action  of  the  punch,  since  annealing, 
which  furnishes  opportunity  for  equalization  of  stress,  restores  the 
strength.  Third,  the  injury  is  only  in  the  immediate  vicinity  of 
the  punched  hole,  since  reaming  out  'lo"  on  a  side  removes  all  the 
injured  material.  In  ordinary  boiler  work  the  plates  are  simply 
punched  and  riveted.  If  better  work  is  required  the  plates  must  be 
drilled,  or  punched  small  and  reamed,  or  punched  and  annealed. 
Drilling  is  slow  and  therefore  expensive  ;  annealing  is  apt  to  change 
the  plates  and  requires  large  expensive  furnaces.  Punching  small 
and  reaming,  is  probably  the  best  method. 

95.  Riveted  Joints  are  of  two  general  kinds  :  First,  Lap  Joints, 
in  which  the  sheets  to  be  joined  are  lapped  upon  each  other  and 
joined  by  a  seam  of  rivets,  as  in  Fig.  104  a.  Second,  Butt  Joints, 
in  which  the  edges  of  the  sheets  abut  against  each  other,  and  a 
strip  called  a  "cover  plate'"  or  "butt  strap"  is  riveted  to  both 
edges  of  the  sheet,  as  in  c. 

There  are  two  kinds  of  riveting :  Single,  in  which  there  is  but 
one  row  of  rivets,  as  in  a  ;  and  double,  where  there  are  two  rows. 


102  MACHINE    DESIGN. 

Double  riveting  is  subdivided  into ''chain  riveting,"  ?;,  and  "  zig- 
zag" or  "staggered"  riveting,  (/. 

Lap  joints  may  be  single,  double  chain,  or  double  staggered 
riveted. 

Butt  joints  m-ay  have  a  single  strap,  as  in  c,  or  double  strap  ; 
i.  e.,  an  exactly  similar  one  is  placed  on  the  other  side  of  the  joint. 
Butt  joints  with  either  single  or  double  strap  may  be  single,  double 
chain,  or  double  staggered  riveted. 

To  sum  up,  there  are  : 

fSingle  Riveted 

Lap  Joints^  Double  Chain  " 

^^       "      Staggered         " 

f  ( Single                      Riveted 

I  Single  Strap<^  Double  Chain  " 

I  i^      "       Staggered         " 
Butt  Joints^ 

I  rSingle                      Riveted 
I  Double      ''    <  Double  Chain 

L  I      "       Staggered 

A  riveted  joint  may  yield  in  any  one  of  four  ways  :  First,  by 
the  rivet  shearing.  Second,  by  the  plate  yielding  to  tension  on  the 
line  AB,  Fig.  105  a.  Third,  by  the  rivet  tearing  out  through  the 
margin,  as  in  c.  Fourth,  the  rivet  and  sheet  bear  upon  each  other 
at  D  and  E  in  d,  and  are  both  in  compression.  If  the  unit  stress 
upon  these  surfaces  becomes  too  great,  the  rivet  is  weakened  to 
resist  shearing,  or  the  plate  to  resist  tension,  and  failure  may  occur. 
This  pressure  of  the  rivet  on  the  sheet  is  called  "  bearing  pressure." 

96.  As  a  preliminary  to  the  designing  of  joints  it  is  necessary  to 
know  the  strength  of  the  rivets  to  resist  shear  ;  of  the  plate  to  resist 
tension  ;  and  of  the  rivets  and  plate  to  resist  bearing  pressure. 
These  values  must  not  be  taken  from  tables  of  the  strength  of 
the  materials  of  which  the  plate  and  rivets  are  made,  but  must  be 
derived  from  experiments  upon  actual  riveted  joints  tested  to  rup- 
ture.    The  reason  for  this  is  that  the  conditions  of  stress  are  modi- 


/y'a/0/. 


Z 


i 


\ 


y 


r/(7./^3. 


K 


\ 


m 


s 


^ 


o 
o 


\ 


cr 


E 


^^ 


OOl 

oo 


\ 

1 

L 


<.f^^^^^^^       ,22ZZ^^^^^^^ 


^itx^ 


v 


0»  THB 


OHIVBHSITT] 


RIVETED    JOINTS. 


lOH 


fied  somewhat  in  the  joint.  For  instance,  in  single  strap  butt 
joints,  and  in  lap  joints,  the  line  of  stress  being  the  centre  line  of 
plates,  and  the  plates  joined  being  offset,  flexure  results  and  the 
plate  is  weaker  to  resist  tension  ;  if  the  joint  yield  to  this  stress  in 
the  slightest  degree  the  "  bearing  pressure  "  is  localized,  and  becomes 
more  destructive.  Extensive  and  accurate  experiments  have  been 
made  upon  actual  joints  and  the  results  are  available  in  Stoney's 
"  Strength  and  Proportions  of  Riveted  Joints."  The  constants 
given  are  taken  from  this  book. 


steel. 


Ultimate  shearing  strength  of  rivets,  single  shear 

double   *'       

Ultimate  tensile  strength  of  plate  between  rivet  holes, 

single  shear 

Ultimate   bearing    pressure    per  sq.  inch  of    diametral 

plane  of  rivet,  single  shear 

Ultimate  bearing  pressure   per   sq.   inch    of   diametral 

plane  of  rivet,  double  shear 


97.  The  theoretical  diameter  of  rivet  for  a  given  thickness  of 
plate  may  now  be  determined.  Let  d  -=  diameter  of  rivet  hole  ;  t  = 
thickness  of  plate;  p  =  pitch  of  rivets;  T=  ultimate  tensile  strength 
of  plate  between  rivet  holes  ;  S  =^  ultimate  shearing  strength  of 
rivets  ;  C  =  ultimate  bearing  pressure. 

The  strength  of  the  rivet  to  resist  shearing  at  AB,  Fig.  106, 
should  be  equal  to  its  strength  to  resist  bearing  pressure  at  A  and  (7, 
and  hence  the  expressions  for  those  strengths  may  be  equated,  thus: 


Solving, 


d^ 


Ctd  =  Sd' 


Ct 


4* 


67000 


S  X  0-7854      40000  X  07854 


2t. 


Hence,  for  equal  strength  to  resist  bearing  pressure  and  shear,  tKe 
diameter  of  the  rivet  should  equal  twice  the  thickness  of  the  plate. 


104 


MACHINE    DESIGN. 


Let  the  results  thus  derived  be  compared  with  the  valaes  that  are 
used  in  actual  practice.     See  table. 


Comparative  Values  in  Inches  of  Rivet  Diameter  for  Different  Values 
OF  Thickness  of  Plate. 


i 

2/ 

i-n/^ 

d 

3x6 

X 

% 

K 

% 

^6 

% 

5,6 

% 

"16 

% 

% 

K 

% 

%-% 

X 

1 

% 

%-% 

% 

m 

'\6 

%-l 

% 

\% 

1^6 

1-lK 

K 

iM 

IK 

1-1% 

1 

2 

1^x6 

1-114 

m 

2.14 

I^-IK 

The  first  column  gives  the  thickness  of  the  plate  ;  the  second  the 
diameter  of  the  rivet  =  2i;  the  third  gives  the  rivet  diameter  calcu- 
lated from  the  formula  of  Professor  Unwin,  d  =  1"2  \/t]  the  fourth 
column  gives  rivet  diameters  as  found  in  practice,  taken  from 
Stoney's  book,  page  12.  rf  —  2^  agrees  with  practice  up  to  %"  plates, 
but  for  thicker  plates  it  gives  values  that  are  too  large.  The  reason 
for  this  is  that  the  difficulty  in  driving  rivets  increases  very  rapidly 
with  their  size  ;  IJ"  or  \%"  being  the  largest  rivet  that  can  be  driven 
conveniently.  The  equality  of  strength  to  resist  bearing  pressure 
and  shear  is  therefore  sacrificed  to  convenience  in  manipulation. 
As  the  diameter  of  the  rivet  is  increased  the  area  to  resist  bearing 
pressure  increases  less  rapidly  than  the  area  to  resist  shear  (the 
thickness  of  the  plate  remaining  the  same),  the  former  varying  as 


RIVETED    JOINTS.  105 

r/,  and  the  latter  as  d^;  therefore  if  r/is  not  increased  as  much  as  is 
necessary  for  equality  of  strength,  the  excess  of  strength  will  be 
to  resist  bearing  pressure.  If  the  other  parts  of  the  joint  are  made 
as  strong  as  the  rivet  in  shear,  and  this  strength  is  calculated  from 
the  stress  to  be  resisted,  the  joint  will  evidently  be  correctly  pro- 
portioned. 

To  calculate  the  diameter  of  rivet  for  a  butt  joint  with  double 
cover  plates.  —  The  rivet  is  in  double  shear,  and  therefore  ultimate 
bearing  pressure  =  89000  lbs.  per  square  inch  =  C.  And  also  ulti- 
mate shear  pressure  =  35000  lbs,  per  square  inch  =  S'. 

liiquating  as  before  Cat  =  — -r —  =    ^    . 

^  ,  .  ,  ,       2Ct       2X89000Xf       ,  ..,  , 

From  which  c^  =  — -  =: o-z^^a/^     =  1'^^  nearly. 

^  r  -  X  ooUUU 

Comparison  of  results  of  this  formula  with  tables  of  dimensions  of 
practice,  shows  them  to  be  too  large.  The  following  empirical  for- 
mulas may  be  trusted  : 

For  thin  plates  —  for  iron  d  =  l"3t;  for  steel,  d  =-- 1*25 1. 
"    thick     "  "        d  =  l-lt;        "         d  =  l'12ot. 

98.  The  next  value  to  be  determined  is  the  pitch  of  the  rivets, 
i.  e.,  the  distance  from  the  centre  of  one  rivet  to  the  centre  of  the 
next  one.  See  Fig.  107.  It  is  required  to  make  the  pitch  of  such  a 
value  that  the  strength  of  the  plate  between  rivet  holes  to  resist, 
tension  shall  equal  the  strength  of  the  rivet  to  resist  shear.  It  has 
already  been  shown  that  the  strength  to  resist  bearing  pressure  is 
equal  to,  or  greater  than,  the  strength  to  resist  shear.  Equate  ex- 
pressions for  shearing  strength  of  the  rivet,  and  tensile  strength  of 
the  plate  on  a  section  through  the  rivet  holes,  and  solve  for  p^ 
pitch.     For  a  single  riveted  lap  joint, 

^^S^Ttip-d). 

,  .  ,  O'lSoWS  -^  Ttd 

h  rom  which  p  = ^ . 

14  It 


106  MACHINE    DESIGN. 

LetS  =  40000  and  T  =  40000. 

Then  if  t  =  i",  d=    V\  P  =  1.28". 

t  =  r,  d=    I";  p  =  1.79". 

t  =  \\  d=    f ;  p  =  2.06". 

t  =  V,  d  =  li"\  p  =  2.12". 

All  of  these  agree  with  Stoney's  "Table  of  Boilermaker's  Propor- 
tions," lap  joints,  iron  plates,  and  rivets,  except  for  t  =  i".  This 
formula  may,  therefore,  be  trusted  except  for  very  thin  plates. 

To  find  p  for  butt  joints  with  double  straps,  single  riveted.  — 
Since  the  rivet  is  in  double  shear, 

2  X  0-7854(^^5"  -|-  Ttd 
P  = Tt ' 

S'  =  35000  lbs.  per  square  inch,  the  value  for  double  shear. 

For  steel  plates  and  steel  rivets,  the  values  of  the  constants,  T 
and  *S',  need  to  be  changed  in  above  formulas.     See  values  given. 

To  find  the  pitch  of  double  riveted  joints  the  method  is  the  same. 
There  are,  however,  two  rivets  now  to  support  the  strip  of  plate 
between  holes,  instead  of  one,  as  in  the  single  joint.  See  Fig.  107. 
Therefore  the  first  formula  for  p,  multiplying  the  shearing  strength 
by  2,  becomes 

I'bld'S+Ttd 
p:^  - 


For  double  shear  p  = 


Tt 
S'Ud'S'  +  Ttd 


Tt 
S'  being  value  for  double  shear. 

99.  The  margin  in  a  riveted  joint  is  the  distance  from  the  edge 
of  the  sheet  to  the  rivet  hole.  This  must  be  made  of  such  value 
that  there  shall  be  safety  against  failure  by  the  rivet  tearing  out. 
There  can  be  no  satisfactory  theoretical  determination  of  this  value  ; 
but  practice  and  experiments  with  actual  joints  show  that  a  joint 
will  not  yield  in  this  way  if  the  margin  be  made  =  d  =  diameter  of 
the  rivet.  The  distance  between  the  centre  lines  of  the  rows  in 
double  chain  riveting  may  be  taken  =  2'6  d  ;  and  in  double  stag- 


RIVETED    JOINTS. 


107 


gered  riveting  may  be  taken  =  l*88(i.  Thus  the  total  width  of  lap 
for  single  riveting  equals  3c/;  in  double  chain  riveting  =  b'bd;  and  in 
double  staggered  riveting  4'88c/.  The  riveted  joints  considered  can- 
not be  as  strong  as  the  unperforated  plates.  The  ratio  of  strength 
of  joint  to  strength  of  plate  is  called  joint  efficiency.  If  the  joint 
were  equally  strong  to  resist  rupture  in  all  possible  ways,  the  joint 
efficiency  would  equal  the  ratio  of  area  of  plate  through  rivet  sec- 
tion, to  the  area  of  unperforated  section.  Results  obtained  in  this 
way  differ  somewhat  from  the  results  of  actual  tests,  and  the  latter 
values  should  be  used.     See  following  tables. 

Relative  Efficiency  of  Iron  Joints. 


Eflficiency 
Percent. 


Original  solid  plate 

Lap  Joint,  single  riveted,  punched.    . . 

"  "  "        drilled 

double     "  

Butt  Joint,  single  cover,  single  riveted 
**  **       double    " 

"  double    **       single      " 

"  **  "       double    " 


100 
45 
50 
60 
45-50 
60 
55 
66 


Relative  Efficiency  of  Steel  Joints. 


Original  solid  plate 

Lap  Joint,  single  riveted,  punched , 

drilled 

"  double      "        punched 

"  *'        drilled 

Butt  Joint,  double  cover,  double  riveted,  punched. 

drilled... 


Efficiency  Per  Cent. 
-Thickness  of  Plates.-. 

H-i 


100 
50 
55 
75 
80 
75 
80 


100 
45 
50 
70 
75 
70 
75 


100 
40 
45 
65 
70 
65 
70 


These  tables  are  from   Stoney's  "  Strength  and   Proportions  of 
Riveted  Joints." 


108  MACHINE    DESIGN. 

100.  The  following  problem  will  serve  to  illustrate  the  design  of 
riveted  joints  for  boilers.  It  is  required  to  design  a  horizontal 
tubular  boiler  48"  diameter  to  carry  a  working  pressure  of  100  lbs. 
per  square  inch.  A  boiler  of  this  type  consists  of  a  cylindrical 
shell  of  wrought  iron  or  steel  plates  made  up  in  length  of  two  or 
more  courses  or  sections.  Each  course  is  made  by  rolling  a  flat 
sheet  into  a  hollow  cylinder  and  joining  its  edges  by  means  of  a 
riveted  joint,  called  the  longitudinal  joint  or  seam.  The  courses  are 
joined  to  each  other  also  by  riveted  joints,  called  circular  joints  or 
seams.  Circular  heads  of  the  same  material  have  a  flange  turned 
all  around  their  circumference,  by  means  of  which  they  are  ri\eted 
to  the  shell.  The  proper  thickness  of  plate  may  be  determined 
from  :  I.  The  diameter  of  shell  =  48".  II.  The  working  steam 
pressure  per  square  inch  =  100  lbs.  III.  The  tensile  strength  of 
the  material  used  ;  let  steel  plates  be  used  of  60000  lbs,  specified 
tensile  strength.  Preliminary  investigation  of  the  conditions  of 
stress  in  the  cross-section  of  material  cut  by  a  plane.  —  I.  Through 
the  axis  ;  II.  At  right  angles  to  the  axis,  of  a  thin  hollow  cylinder  ; 
the  stress  being  due  to  the  excess  of  internal  pressure  per  square 
inch  over  the  external  pressure  per  square  inch.  Let  l  =  ihe  length 
of  .the  cylindrical  shell  in  inches;  d^the  diameter  of  the  cylin- 
drical shell  in  inches;  j9  =  the  excess  of  internal  over  external 
pressure  per  square  inch  ;  p^-=\init  stress  in  a  longitudinal  section 
of  material  of  the  shell  due  to  ^;  p.^  =  unit  stress  in  a  circular  sec- 
tion of  material  of  the  shell  due  to  p;  ^  =  thickness  of  plate  ;  T= 
ultimate  tensile  strength  of  plate. 

In  a  longitudinal  section  the  stress  =  Idp,  and  the  area  of  metal 

sustaining  it  ~  2lt.     Then  p^  —  -^. 

In  a  circular  section  the  stress  =  —r^,  and  the  area  =  7:dt  nearly. 

Then  «  -  !^  V  J-  _  ^i£ 

Therefore  the  stress  in  the  first  case  is  twice  as  great  as  in  the 
second  ;  and  a  thin  hollow  cylinder  is  twice  as  strong  to  resist  rup- 


RIVETED   JOINTS.  109 

ture  on  a  circular  section  as  on  a  longitudinal  one.  The  latter 
only,  therefore,  need  be  considered  in  determining  ihe  thickness  of 
plate.  Equating  the  stress  due  to  j9  in  a  longitudinal  section  and 
the  strength  of  the  cross-section  of  plate  that  sustains  it,  we  have 

(I'D 

ldp  =  2ltT.     Therefore  i  =  ;r^  =  the  thickness  of  plate  that  would 

just  yield  to  the  unit  pressure  p.  To  get  safe  thickness,  a  factor  of 
safety  must  be  used.  It  is  usually  equal  in  boiler  shells  to  4,  5,  or 
6.  Its  value  is  small  because  the  material  is  highly  resilient  and 
the  changes  of  pressure  are  gradual,  i.  e.,  there  are  no  shocks.  This 
takes  no  account  of  the  riveted  joint,  which  is  the  weakest  longi- 
tudinal section,  e  times  as  strong  as  the  solid  plate ;  e  being  the 
\o\ni  effieiency^=^  01b  if  the  joint  be  double  riveted.     The  formula 

then  becomes  t  =  "^^ .     Substituting  values 

^  6  X  48  X  100  ^,^^„ 

^=2X60000X0-75^^^^'^"^  ''" 

The  circular  joints  will  be  single  riveted  and  joint  efficiency  will  =^ 
0'50.  But  the  stress  is  only  one-half  as  great  as  in  the  longitudinal 
joint,  and  therefore  it  is  stronger  in  the  proportion  0*50  X  2  to 
075  =  1  to  0*75.  From  this  it  is  seen  that  a  circular  joint  whose 
efficiency  is  0'50  is  as  strong  as  the  solid  plate  in  a  longitudinal 
section.  From  the  value  of  t  the  joints  may  now  be  designed. 
Diameter  of  rivet  =  d  =  V2y't  =  l-2i/ 0-3125  =  0-672",  say  0-687"  = 
"i6".     The  pitch  for  a  single  riveted  joint  = 

0-7854^'^5^  4-  Ttd 


^  Tt 

But  (Z  =  "16  =  687";  ^-=50000  for  steel;  r=  60000  for  steel;  1  = 
5i6  =  0-3125.  Substituting  these  values  p  —  1-42".  For  double 
riveted  joint 

j9  — jjT "^  (substituting  as  above)  2-66  . 


110  MACHINE    DESIGN. 

The  margin  =  (i  =  0*687"  =  "16".  The  longitudinal  joint  will  be 
staggered  riveted  and  the  distance  between  rows  =^  I'SSri  =  1*29"  = 
say  1  Sjg".  The  total  lap  in  the  longitudinal  joint  =  4*88c?  =:  3*35". 
The  total  lap  in  the  circular  joint  =  3(Z--— 2  ^e"-  The  joints  are 
therefore  completely  determined,  and  a  detail  of  each,  giving  dimen- 
sions, may  be  drawn  for  the  use  of  the  workmen  who  make  the  tem- 
plets and  lay  out  the  sheets. 


CHAPTER   X. 


DESIGN    OF    JOURNALS. 


101.  Journals  and  the  bearings  or  boxes  with  which  they  engage 
are  the  elements  used  to  constrain  motion  of  rotation  or  vibration 
about  axes  in  machines.  Journals  are  usually  cylindrical,  but  may 
be  conical,  or,  in  rare  cases,  spherical.  The  design  of  journals,  as 
far  as  size  is  concerned,  is  dictated  by  one  or  all  of  the  three  follow- 
ing considerations  :  I.  To  provide  for  safety  against  rupture  or 
excessive  yielding  under  the  applied  forces.  II.  To  provide  for 
maintenance  of  form.  III.  To  provide  against  the  squeezing  out 
of  the  lubricant.  To  illustrate  I.  —  Let  Fig.  108  represent  a  pulley 
on  the  end  of  an  overhanging  shaft,  driven  by  a  belt,  ABC.  Rota- 
tion is  as  indicated  by  the  arrow,  and  the  belt  tensions  are  T^  and 
T.^.  The  journal,  /,  engages  with  a  box  or  bearing,  D.  The  follow- 
ing stresses  are  induced  in  the  journal  :  Torsion,  measured  by  the 
torsional  moment  {T^~  T.^)r.  Flexure,  measured  by  the  bending 
moment  (7",  -f  T.^)a.  This  assumes  a  rigid  shaft  or  a  self-adjusting 
box.  Shear,  resulting  from  the  force  T^  -\-  T.^.  This  journal  must 
therefore  be  so  designed  that  rupture  or  undue  yielding  shall  not 
result  from  any  one  of  these  stresses.  To  illustrate  II.  —  Consider 
the  spindle  journals  of  a  grinding  lathe.  The  forces  applied  are 
very  small ;  but  the  form  of  the  journals  must  be  maintained  to 
insure  accuracy  in  the  product  of  the  machine.  A  relatively  large 
wearing  surface  is  therefore  necessary.  To  illustrate  III.  —  The  pres- 
sure upon  a  journal  resulting  from  the  applied  forces  may  be 
sufficiently  great  to  squeeze  out  the  lubricant.  Metallic  contact, 
heating,  and  abrasion  of  the  surfaces  would  result.     In  what  fol- 


112  MACHINj:    DESIGN. 

lows,  the  area  of  a  journal  means  its  projected  area  ;  i.  e.,  its  length 
X  its  diameter. 

The  allowable  pressure  per  square  inch  of  area  of  a  journal 
varies  with  several  conditions.  To  make  this  clear,  suppose  a  drop 
of  oil  to  be  put  in  the  middle  of  an  accurately  finished  surface 
plate ;  suppose  another  exactly  similar  plate  to  be  placed  upon  it 
for  an  instant ;  the  oil  drop  will  be  spread  out  because  of  the  force 
due  to  the  weight  of  the  upper  plate.  If  the  plate  were  allowed  to 
remain  a  longer  time,  the  oil  would  be  still  further  spread  out,  and 
if  its  weight  and  the  time  were  sufficient,  the  oil  would  finally  be 
entirely  squeezed  out  from  between  the  plates,  and  the  metal  sur- 
faces would  come  in  contact.  The'  squeezing  out  of  the  oil  from 
between  the  rubbing  surfaces  of  a  journal  and  its  box  is,  therefore, 
a  function  of  the  time  as  well  as  of  pressure.  If  the  surfaces  under 
pressure  move  over  each  other,  the  removal  of  the  oil  is  facilitated. 
The  greater  the  velocity  of  movement,  the  more  rapidly  will  the  oil 
be  removed,  and  therefore  the  squeezing  out  of  the  oil  is  also  a 
function  of  the  velocity  of  rubbing  surfaces. 

When  a  journal  is  subjected  to  continuous  pressure  in  one  direc- 
tion, as  for  instance  in  a  shaft  with  a  constant  belt  pull,  or  with  a 
heavy  fly-wheel  upon  it,  this  pressure  has  sufficient  time  to  act,  and 
is  therefore  effective  for  the  removal  of  the  oil.  But  if  the  direction 
of  the  pressure  is  periodically  reversed,  as  in  the  crank-pin  of  a 
steam  engine,  the  time  of  action  is  less,  the  tendency  to  remove  the 
oil  is  reduced,  and  the  oil  has  opportunity  to  return  between  the 
surfaces.  Hence,  a  higher  pressure  per  square  inch  of  journal 
would  be  allow^able  in  the  second  case  than  in  the  first. 

If  the  direction  of  motion  is  also  reversed,  as  in  the  cross  head 
pin  of  a  steam  engine,  the  oil  not  only  has  an  opportunity  to  return 
between  the  surfaces,  but  is  assisted  in  doing  so  by  the  reversed 
motion.  Therefore,  a  still  higher  pressure  per  square  inch  of  jour- 
nal is  allowable.  Practical  experience  bears  out  these  conclusions. 
Thus  in  journals  with  the  direction  of  pressure  constant,  it  is  found 
that  with  ordinary  conditions  of  lubrication  the  heating  and  "  seiz- 
ing "  or  "cutting"  occur  quickly  if  the  pressure  per  square  inch  of 


w 


-prfchi 


G) 


0)00) 


piich 


c 


up  o  o 
®  ®  o 


f/a.  /06 


DESIGN    OF    JOURNALS.  113 

journal  exceed  about  380  lbs.*  But  in  the  crank-pins  of  punching 
machines,  where  the  pressure  acts  for  an  instant,  with  quite  an 
interval  of  rest,  and  where  the  velocity  of  rubbing  surface  is  very 
low  indeed,  the  pressure  is  often  as  high  as  from  2000  to  3000  lbs., 
per  square  inch,  and  there  is  no  tendency  to  heating  or  abrasion. 
In  engine  crank-pins  the  pressure  may  be  from  400  to  800  lbs. 
depending  on  the  velocity  of  rubbing  surface,  and  in  cross-head 
pins  where  the  velocity  is  always  low  it  may  be  from  600  to  1000 
lbs.  The  value  to  be  used  in  each  particular  case  must  be  decided 
by  the  judgment  of  the  designer. 

But  even  if  the  conditions  are  such  that  the  lubricant  is  retained 
between  the  rubbing  surfaces,  heating  may  occur.  There  is  always 
a  frictional  resistance  at  the  surface  of  the  journal  ;  this  resistance 
may  be  reduced  :  a,  by  insuring  accuracy  of  form  and  perfection  of 
surface  in  the  journal  and  its  bearings  ;  b,  by  insuring  that  the 
journal  and  its  bearing  are  in  contact,  except  for  the  film  of  oil, 
throughout  their  entire  surface,  by  means  of  rigidity  of  framing  or 
self-adjusting  boxes,  as  the  case  may  demand  ;  c,  by  selecting  a  suit- 
able lubricant  to  meet  the  conditions,  and  maintaining  the  supply  to 
the  bearing  surfaces.  By  these  means  the  friction  may  be  reduced 
to  a  very  low  value,  but  it  cannot  be  reduced  to  zero. 

There  must  be  some  frictional  resistance,  and  it  is  always  con- 
verting mechanical  energy  into  heat.  This  heat  raises  the  tempera- 
ture of  the  journal  and  its  bearing.  If  the  heat  thus  generated  is 
conducted  and  radiated  away  as  fast  as  it  is  generated,  the  box 
remains  at  a  constant  low  temperature.  If,  however,  the  heat  is 
generated  faster  than  it  can  be  disposed  of,  the  temperature  of  the 
box  rises  till  its  capacity  to  radiate  heat  is  increased  by  the  in- 
creased difference  of  temperature  of  the  box  and  the  surrounding 
air,  so  that  it  is  able  to  dispose  of  the  heat  as  fast  as  it  is  generated. 
This  temperature,  necessary  to  establish  the  equilibrium  of  heat 
generation  and  disposal,  may  under  certain  conditions  be  high 
enough  to  destroy  the  lubricant,  or  even  to  melt  out  a  babbitt  metal 

*See  Mr.  Tower's  experiments  in  the  **  Minutes  of  the  Institution  of 
Mechanical  Engineers." 

15 


114  MACHINE    DESIGN. 

box  lining.  Suppose  now  that  a  journal  is  running  under  certain 
conditions  of  pressure  and  surface  velocity,  and  that  it  remains 
entirely  cool.  Suppose  next  that  while  all  other  conditions  are 
kept  exactly  the  same,  the  velocity  is  increased.  All  modern 
experiments  on  the  friction  in  journals  show  that  the  friction  in- 
creases with  the  increase  of  the  velocity  of  rubbing  surface.  Therefore 
the  increase  in  velocity  would  increase  the  frictional  resistance  at 
the  surface  of  the  journal,  and  the  space  through  which  this  resist- 
ance acts  would  be  greater  in  proportion  to  the  increase  in  velocity. 
The  work  of  the  friction  at  the  surface  of  the  journal  is  therefore 
increased,  because  both  the  force  and  the  space  factors  are  increased. 
It  is  this  work  of  friction  which  has  been  so  increased,  that  produces 
the  heat  that  tends  to  raise  the  temperature  of  the  journal  and  its 
box.  The  rate  of  generation  of  heat  has  therefore  been  increased 
by  the  increase  in  velocity,  but  the  box  has  not  been  changed  in 
any  way  and  therefore  its  capacity  for  disposing  of  heat  is  the  same 
as  it  was  before,  and  hence  the  tendency  of  the  journal  and  its 
bearing  to  heat  is  greater  than  it  was  before  the  increase  in  velocity. 
Some  change  in  the  proportions  of  the  journal  must  be  made  in 
order  to  keep  the  tendency  to  heat  the  same  as  it  was  before  the 
increase  in  velocity.  If  the  diameter  of  the  journal  be  increased, 
the  radiating  surface  of  the  box  will  be  proportionately  increased. 
But  the  space  factor  of  the  friction  will  be.  increased  in  the  same 
proportion,  and  therefore  it  will  be  apparent  that  this  change  has 
not  affected  the  relation  of  the  rate  of  generation  of  heat  to  the  dis- 
posal of  it.  But  if  the  length  of  the  journal  be  increased,  the  work 
of  friction  is  the  same  as  before  and  the  radiating  surface  of  the. 
box  is  increased  and  the  tendency  of  the  box  to  heat  is  reduced.  If, 
therefore,  the  conditions  are  such  that  the  tendency  to  heat  in  a 
journal,  because  of  the  work  of  the  friction  at  its  surface,  is  the 
vital  point  in  design,  it  will  be  clear  that  the  length  of  the  journal 
is  dictated  by  it,  but  not  the  diameter.  The  reason  why  high  speed 
journals  have  greater  length  in  proportion  to  their  diameter  than 
low  speed  journals  will  now  be  apparent, 

102.  Problem.  —  To   design   the  main  journal   of   a   side   crank 


DESIGN    OF    JOURNALS.  115 

engine. — The  data  are  as  follows:  Diameter  of  steam  cylinder  = 
16" ;  boiler  pressure  =  100  lbs.  per  sq.  in.  by  gauge.  Then  the 
maximum  force  upon  the  steam  piston  due  to  steam  pressure  = 
100  X  8"^7r  =  20106  lbs.  Suppose  that  the  least  expensive  stress 
member  is  a  "  breaking  piece,"  i.  e.,  it  will  yield  and  relieve  the 
stress  in  the  other  stress  members  when  the  total  applied  force  ^= 
80000  lbs.,  about  four  times  the  maximum  working  force.  In 
Fig.  109,  DE  is  the  centre  line  of  the  engine  ;  C  is  the  crank- 
pin  ;  A  is  the  crank  disc,  and  B  is  the  journal  to  be  designed. 
The  force  P,  =  80000  lbs.,  is  the  greatest  force  that  can  act  in 
the  line  DE.  The  journal  is  supported  up  to  the  line  FG. 
In  the  section  HK  there  is  flexure  stress  measured  by  the  flex- 
ure moment  PL  I  in  this  case  =  6".  The  breaking  piece  only 
yields  when  the  crank  is  at  or  near  its  centre  ;  hence,  the 
torsional  stress  may  be  neglected.  The  radius  of  the  shaft  for 
safety  against  the  moment  PI,  may  be   found  from  the  formula 

SI  Pic  7tr* 

Pl=  —  ;    from   which  1=  -—  .     But  /  for  circular  section  =  — -  : 
c  S  4 

4Pl 
and  c  =  r.      Hence  r^  =  —^.     Let  S  for  machinery  steel  =  12000. 

This  gives  a   factor   of   safety  =  T97^7^7J  =  5.     Substituting   values, 

P=  80000,  I  =-■  6",  and  S  =  12000,  in  the  above  equation,  gives  r  = 
8'71",  say  8|".  Hence,  the  shaft  diameter  =  7^".  This  value  de- 
pends upon  the  assumptions  made  for  P,  the  strength  of  the  break- 
ing piece,  and  for  S,  the  safe  stress  for  the  material  used.  Different 
values  might  have  been  assumed,  and  would,  of  course,  have  given 
different  results.  The  length  of  such  a  journal  is  determined  by 
practical  considerations.  In  this  case  the  length  should  be  about 
twice  the  diameter  =  15",  in  order  that  convenient  means  may  be 
supplied  for  taking  up  wear.  The  projected  area  of  journal  =  7*5" 
X  15  =  112'5  square  inches.  Assume  350  lbs.  safe  pressure  per 
square  inch  of  journal.  This  would  admit  of  a  working  pressure 
of  350  X  112*5  =  39375.  It  is  evident  without  investigation  that 
this  is  greater  than  any  working  load  for  this  journal. 


116  MACHINE   DESIGN. 

103.  Problem.  —  To  design  the   crank-pin   for  the  same  engine. 

—  The  bending  moment  now  equals  Pl^.     Assume  l^  =  3".     Then 

.,      4  X  80000  X  3   ,  ,  .  ,  ,-,  ... ,  .^„      ^,       ,         , 

r*  = — ,  from  which  r  --=  294,  say  3  .     Therefore,  d  = 

TT  X   J-  ^UUU 

6";  and  since  the  assumed  length  =  6",  the  journal  area  =  36  sq.  in. 
Then  if  the  allowable  pressure  per  square  inch  =  700  lbs.,  the  total 
allowable  working  pressure  —  25200  lbs.  This  is  greater  than  the 
possible  working  pressure,  and  hence  the  lubricant  would  not  be 
squeezed  out.  The  size  of  both  journal  and  crank-pin  is  therefore 
dictated  by  the  maximum  bending  moment. 

104.  To  design  the  cross-head  pin  for  the  same  engine.  —  In  Fig. 
110,  C  represents  the  cross-head  pin.  The  force  P,  =  80000,  may 
be  applied  as  indicated.  The  pin  is  supported  at  both  ends,  and 
the  connecting-rod  box  bears  upon  it  throughout  its  entire  length, 
AD  or  BE.  The  pin  would  yield  by  shearing  on  the  sections  AB 
and  DE.  The  shearing  strength  of  the  machinery  steel,  of  which  it 
would  be  made,  may  be  assumed  equal  to  50000  lbs.  A  stress  of 
8000  lbs.  would  therefore  give  a  factor  of  safety  of  6  -|--  The  neces- 
sary area  in  shear  would  equal         „  =  10,  or  5  square  inches  for 

each  section.  This  corresponds  to  a  diameter  of  2*5  -f-.  The  length 
of  pin  may  be  found  as  follows  :  Find  the  mean  working  force  upon 
the  pin,  by  drawing  the  indicator  card,  as  modified  by  acceleration 
of  reciprocating  parts,  and  multiplying  the  value  of  its  mean 
ordinate,  in  pounds  per  square  inch,  by  the  piston  area.  The  value 
for  this  case  =  about  12000  lbs.  The  allowable  pressure  per  square 
inch  of  journal  =  800  lbs.  Hence  the  journal  area  =  12000  -^  800 
=  15.  The  length  then  =  15  -f-  2-5  =  6".  The  diameter  of  the 
cross-head  pin,  therefore,  is  dictated  by  the  applied  force,  while  its 
length  depends  upon  the  maintenance  of  lubrication.  The  judg- 
ment of  the  designer  might  require  this  pin  to  be  still  larger  to  re- 
duce wear  and  to  maintain  its  form. 

Journals  whose  maintenance  of  form  is  of  chief  importance, 
must  be  designed  from  precedent,  or  according  to  the  judgment  of 
the  designer.     No  theory  can  lead  to  correct  proportions.     In  fact 


DESIGN    OF    JOURNALS.  117 

these   proportions   are   eventually   determined   by   the   process   of 
Machine  Evolution. 

105.  Thrust  Journals.  —  When  a  rotating  machine  part  is  sub- 
jected to  pressure  parallel  to  the  axis  of  rotation,  means  must  be 
provided  for  the  safe  resistance  of  that  pressure.  In  the  case  of 
vertical  shafts  the  pressure  is  due  to  the  weight  of  the  shaft  and  its 
attached  parts  —  as  the  shafts  of  turbine  water-wheels  that  rotate 
about  vertical  axes.  In  other  cases  the  pressure  is  due  to  the  work- 
ing force  —  as  the  shafts  of  propeller  wheels,  the  spindles  of  a  chuck- 
ing lathe,  etc.  The  end  thrust  of  vertical  shafts  is  very  often 
resisted  by  the  "squared  up"  end  of  the  shaft.  This  is  in- 
serted in  a  bronze  or  brass  "  bush,"  which  embraces  it  to 
prevent  lateral  motion,  as  in  Fig.  111.  If  the  pressure  be  too 
great,  the  end  of  the  shaft  may  be  enlarged  so  as  to  increase  the 
bearing  surface,  thereby  reducing  the  pressure  per  square  inch. 
This  enlargement  must  be  within  narrow  limits,  however.  See  Fig. 
112.  AB  is  the  axis  of  rotation,  and  ACD  is  the  rotating  part,  its 
bearing  being  enlarged  at  CD.  Let  the  conditions  of  wear  be  con- 
sidered. The  velocity  of  rubbing  surface  varies  from  zero  at  the 
axis  to  a  maximum  at  C  and  D.  It  has  been  seen  that  the  increase 
of  the  velocity  of  rubbing  surface  increases  both  the  force  of  the 
friction  and  the  space,  through  which  that  force  acts ;  it  therefore 
increases  the  work  of  the  friction,  and  therefore  the  tendency  to 
wear.  From  this  it  will  be  seen  that  the  tendency  to  wear  increases 
from  the  centre  to  the  circumference  of  this  "radial  bearing,"  and 
that,  after  the  bearing  has  run  for  a  while,  the  pressure  will  be 
localized  near  the  centre,  and  heating  and  abrasion  may  result. 
For  this  reason,  where  there  is  severe  stress  to  be  resisted,  the  bear- 
ing is  usually  divided  up  into  several  parts,  the  result  being  what 
is  known  as  a  "collar  thrust  bearing,"  as  shown  in  Fig.  113.  By 
the  increase  in  the  number  of  collars,  the  bearing  surface  may  be 
increased  without  increasing  the  tendency  to  unequal  wear.  The 
radial  dimension  of  the  bearing  is  kept  as  small  as  is  consistent 
with  the  other  considerations  of  the  design.  It  is  found  that  the 
"  tractrix,"  the  curve  of  constant  tangent,  gives  the  same  work  of 


118  MACHINE    DESIGN. 

friction,  and  hence  the  same  tendency  to  wear  in  the  direction  of 
the  axis  of  rotation,  for  all  parts  of  the  wearing  surface.  (See 
'^Church's  Mechanics,"  page  181.)  This  is  without  doubt  the  best 
form  for  a  thrust  bearing,  but  the  difficulties  in  the  way  of  the 
accurate  production  of  its  curved  outlines  have  interfered  with  its 
extensive  use. 

The  pressure  that  is  allowable  per  square  inch  of  projected  area 
of  the  bearing  surface  varies  in  thrust  bearings  with  several  con- 
ditions, as  it  does  in  journals  subjected  to  pressure  at  right  angles 
to  the  axis.  Thus  in  the  pivots  of  turn-tables,  swing  bridges, 
cranes,  and  the  like,  the  movement  is  slow  and  never  continuous, 
often  being  reversed;  and  also  the  conditions  are  such  that  "bath 
lubrication  "  may  be  used,  and  the  allowable  unit  pressure  is  very 
high  —  equal  often  to  1500  pounds  per  square  inch,  and  in  some 
cases  greatly  exceeding  that  value.  The  following  table  may  be 
used  as  an  approximate  guide  in  the  designing  of  thrust  bearings. 
The  material  of  the  thrust  journal  is  wrought  iron  or  steel,  and 
the  bearing  is  of  bronze  or  brass  (babbitt  metal  is  seldom  used  for 
this  purpose).  Bath  lubrication  is  used,  i.  e.,  the  running  surfaces 
are  submerged  constantly  in  a  bath  of  oil. 

Mean  Velocity  Allowable  Unit  Pressure, 
of     rubbing  lbs.  per  square  inch 

surface,  feet  of  projected  area  of 

per  minute.  the  rubbing  surface. 

Up  to  50 1000 

50  to  100            600 

100  to  150            350 

150  to  200            100 

Above  200                      50 

If  the  journal  is  of  cast  iron  and  runs  on  bronze  or  brass,  the 
values  of  allowable  pressure  given  should  be  divided  by  2. 

106.  Examples  to  illustrate  the  design  of  thrust  journals. 

Example  I.  — It  is  required  to  design  a  thrust  journal  whose  out- 
line is  a  tractrix.  It  is  required  to  support  a  vertical  shaft  which  with 
its  attached  parts  weighs  2000  lbs.,  and  runs  at  a  rotative  speed  of  200 
revolutions  per  minute.     The  dimensions  of  the  thrust  journal  are 


ED 


qf^\jU3 


/y.  f/^. 


fuiriVBRSITYJ 


DESIGN    OF    JOURNALS.  119 

as  yet  unknown,  and  therefore  the  velocity  of  rubbing  surface  must 
be  estimated.  Suppose  that  the  mean  diameter  of  the  journal  is  2"; 
then  the  mean  velocity  of  rubbing  surface  will  he  2  X  t:  X  N -~  12 
=  103  feet  per  minute.  This  is  so  near  the  limit  in  the  table 
between  an  allowable  pressure  of  350  and  600  that  an  intermediate 
value  may  be  used,  say  450  pounds.  The  projected  area  of  the 
journal  then  will  equal  the  total  pressure  divided  by  the  allowable 
pressure  per  square  inch  of  the  journal  ^==2000 -^450  =  4*44  square 
inches.  The  journal  must  not  be  pointed,  as  in  (a)  Fig.  114,  but 
must  be  as  shown  in  (h).  The  dimension  BC  may  be  assumed 
equal  to  1".  The  projected  area  of  the  journal  is  equal  to  the  cir- 
cular area  whose  diameter  is  AD,  minus  the  circular  area  whose 
diameter  is  BC,  and  this  may  be  equated  with  the  required  value, 
equal  4'44,  and  the  equation  solved  for  the  required  dimension,  AD. 

Thus  -^ — r^ -5^ — j^—  =  4-4:4: 

4  4 

4-44.  V  4 
Therefore  (ADy  =  ^     -|-  (BCy. 


AD=i.^Q'6S  =  2'5S". 

In  order  now  to  draw  the  required  journal,  lay  off  from  the  axis 
EF  the  distance  EG,  equal  half  AD,  and  through  the  point  G 
draw  a  tractrix  whose  constant  tangent  is  equal  to  EG,  con- 
tinuing the  curve  till  it  reaches  a  point  C,  such  that  FC  is  equal  to 
half  the  assumed  value  of  BC.  The  vertical  dimension  of  the  jour- 
nal is  thereby  determined,  and  the  corresponding  curve,  BH,  may 
be  drawn  on  the  other  side  of  the  axis  EF. 

107.  Example  11.  —  It  is  required  to  design  the  collar  thrust  jour- 
nal that  is  to  receive  the  propelling  pressure  from  the  screw  of  a  small 
yacht.  The  necessary  data  are  as  follows  :  The  maximum  power 
delivered  to  the  shaft  is  70  H.  P.;  pitch  of  screw  is  4  feet :  slip  of 
screw  is  20,% ;  shaft  revolves  250  times  per  minute  ;  diameter  of 
shaft  is  4". 

For  evety  revolution   of  the  screw  the  yacht  moves  forward  a 


120  MACHINE    DESIGN. 

distance  =  4  ft.  less  20%  =S2  ft.,  and  the  speed  of  the  yacht  in  feet 
per  minute  ==  250  X  3-2  ^  800.  70  H.  P.  =  70  X  33000  =  2,310,000 
foot-pounds  per  minute.  This  work  may  be  resolved  into  its  factors 
of  force  and  space,  and  the  propelling  force  is  equal  to  2,310,000 -f- 
800  =  2900  lbs.  nearly.  The  shaft  is  4"  diameter,  and  the  collars 
must  project  beyond  its  surface.  Estimate  that  the  mean  radius  of 
the  rubbing  surface  is  4'5",  then  the  mean  velocity  of  rubbing  sur- 
face would  equal  4*5X7^-^12X  250=294  feet  per  minute.  The 
allowable  value  of  pressure  per  square  inch  of  journal  surface  for  a 
velocity  above  200  ft.  per  minute  is  50  lbs.  The  necessary  area  of 
the  journal  surface  is  therefore  =  2900  ^- 50  =^  58  square  inches. 
It  has  been  seen  that  it  is  desirable  to  keep  the  radial  dimension  of 
the  collar  surface  as  small  as  possible  in  order  to  have  as  nearly 
the  same  velocity  at  all  parts  of  the  rubbing  surface  as  possible. 
The  width  of  collar  in  this  case  will  be  assumed  =:0'75";  then  the 
bearing  surface  in  each  collar 

=  ^'^'J  "  -  ^^  =  23-7  -  12-5  =  11-2. 
4  4 

Then  the  number  of  collars  equals  the  total  required  area  divided  by 
the  area  of  each  collar  =  58  h-  11*2  =  5' 18,  say  6. 

108.  Bearings  and  Boxes.  —  The  function  of  a  bearing  or  box  is 
to  insure  that  the  journal  with  which  it  engages  shall  have  an 
accurate  motion  of  rotation  or  vibration  about  the  given  axis.  It 
must  therefore  fit  the  journal  without  lost  motion  ;  must  afford 
means  of  taking  up  the  lost  motion  that  results  necessarily  from 
wear ;  must  resist  the  forces  that  come  upon  it  through  the  jour- 
nal, without  undue  yielding  ;  must  have  the  wearing  surface  of  such 
material  as  will  run  in  contact  with  the  material  of  the  journal 
with  the  least  possible  friction,  and  least  tendency  to  heating  and 
abrasion  ;  and  must  usually  include  some  device  for  the  main- 
tenance of  the  lubrication.  The  selection  of  the  materials  and  the 
providing  of  sufficient  strength  and  stiffness  depends  upon  prin- 
ciples already  considered,  and  so  it  remains  to  discuss  the  means 
for  the  taking  up  of  necessary  wear  and  for  providing  lubrication. 


DESIGN    OF    JOURNALS.  121 

Boxes  are  sometimes  made  solid  rings  or  shells,  the  journal  beino; 
inserted  endwise.  In  this  case  the  wear  can  only  be  taken  up  by 
making  the  engaging  surfaces  of  the  box  and  journal  conical,  and 
providing  for  endwise  adjustment  either  of  the  box  itself  or  of  the 
part  carrying  the  journal.  Thus,  in  Fig.  115,  the  collars  for  the 
preventing  of  end  motion  while  running,  are  jamb  nuts,  and  loose- 
ness between  the  journal  and  box  may  be  taken  up  by  moving  the 
journal  axially  toward  the  left. 

By  far  the  greater  number  of  boxes,  however,  are  made  in  sec- 
tions, and  the  lost  motion  is  taken  up  by  moving  one  or  more 
sections  toward  the  axis  of  rotation.  The  tendency  to  wear  is 
usually  in  one  direction,  and  it  is  sufficient  to  divide  the  box 
into  halves.  Thus,  in  Fig.  116,  the  journal  rotates  about  the 
axis  0,  and  all  the  wear  is  due  to  the  pressure  P  acting  in  the 
direction  shown.  The  wear  will  therefore  be  at  the  bottom  of 
the  box.  It  will  suffice  for  the  taking  up  of  wear  to  dress  off  the 
surfaces  at  aa,  and  thus  the  box  cap  may  be  drawn  further  down 
by  the  bolts,  and  the  lost  motion  is  reduced  to  an  admissible  value. 
•'Liners"  or  "shims,"  which  are  thin  pieces  of  sheet  metal,  may  be 
inserted  between  the  surfaces  of  division  of  the  box  at  aa^  and  may 
be  removed  successively  for  the  lowering  of  the  box  cap  as  the  wear 
renders  it  necessary.  If  the  axis  of  the  journal  must  be  kept  in  a 
constant  position,  the  lower  half  of  the  box  must  be  capable  of  being 
raised. 

Sometimes,  as  in  the  case  of  the  box  for  the  main  journal  of 
a  steam  engine  shaft,  the  direction  of  wear  is  not  constant.  Thus, 
in  Fig.  117,  A  represents  the  main  shaft  of  an  engine.  There  is  a 
tendency  to  wear  in  the  direction  B  because  of  the  weight  of  the 
shaft  and  its  attached  parts  ;  there  is  also  a  tendency  to  wear 
because  of  the  pressure  that  comes  through  the  connecting-rod  and 
crank.  The  direction  of  this  pressure  is  constantly  varying,  but 
the  average  direction  on  forward  and  return  stroke  may  be  repre- 
sented by  C  and  D.  Provision  needs  to  be  made,  therefore,  for 
taking  up  wear  in  these  two  directions.  If  the  box  be  divided  on 
the  line  EF^  wear  will  be  taken  up  vertically  and  horizontally  by 

i6 


122  MACHINE  design: 

reducing  the  liners.  Usually,  however,  in  the  larger  engines  the 
box  is  divided  into  four  sections,  A,  B,  C,  and  D  (Fig.  118),  and  A 
and  C  are  capable  of  being  moved  toward  the  shaft  by  means  of 
screws  or  wedges,  while  D  may  be  raised  by  the  insertion  of 
"shims." 

The  lost  motion  between  a  journal  and  its  box  is  sometimes 
taken  up  by  making  the  box  as  shown  in  Fig.  119.  The  external 
surface  of  the  box  is  conical  and  fits  in  a  conical  hole  in  the 
machine  frame.  The  box  is  split  entirely  through  at  A,  parallel  to 
the  axis,  and  partly  through  at  B  and  C.  The  ends  of  the  box  are 
threaded,  and  the  nuts  E  and  F  are  screwed  on.  After  the  journal 
has  run  long  enough  so  that  there  is  an  unallowable  amount  of  lost 
motion,  the  nut  F  is  loosened  and  E  is  screwed  up  ;  the  effect  being 
to  draw  the  conical  box  further  into  the  conical  hole  in  the  machine 
frame ;  the  hole  through  the  box  is  thereby  closed  up,  and 
lost  motion  is  reduced.  After  this  operation  the  hole  cannot  be 
truly  cylindrical,  and  if  the  cylindrical  form  of  the  journal  has 
been  maintained,  it  will  not  have  a  bearing  throughout  its  entire 
surface.  This  is  not  usually  of  very  great  importance,  however, 
and  the  form  of  box  has  the  advantage  that  it  holds  the  axis  of  the 
journal  in  a  constant  position. 

All  boxes  in  self-contained  machines,  like  engines  or  machine 
tools,  need  to  be  rigidly  supported  to  prevent  the  localization  of 
pressure,  since  the  parts  that  carry  the  journals  are  made  as  rigid 
as  possible.  In  line  shafts  and  other  parts  carrying  journals,  when 
the  length  is  great  in  comparison  to  the  lateral  dimensions,  some 
yielding  must  necessarily  occur,  and  if  the  boxes  were  rigid,  local- 
ization of  pressure  would  result.  Hence  "self-adjusting  boxes" 
are  used.  A  point  in  the  axis  of  rotation  at  the  centre  of  the 
length  of  the  box  is  held  immovable,  but  the  box  is  free  to  move 
in  any  way  about  this  point,  and  thus  adjusts  itself  to  any  yield- 
ing of  the  shaft.  This  result  is  attained  as  shown  in  Fig.  120.  0 
is  the  centre  of  the  motion  of  the  box  ;  B  and  A  are  spherical  sur- 
faces formed  on  the  box,  their  centre  being  at  0.  The  support  for  the 
box  carries  internal  spherical  surfaces  which  engage  with  A  and  B. 


r,0^   Of  Tim       ^ 


DESia.N.   OF    JOURNALS.  128 

Thus  the  point  0  is  always  held  in  a  constant  position,  but  the  box 
itself  is  free  to  move  in  any  way  about  0  as  a  centre.  Therefore 
the  box  adjusts  itself,  within  limits,  to  any  position  of  the  shaft, 
and  hence  the  localization  of  pressure  is  impossible. 

In  thrust  bearings  for  vertical  shafts  the  weight  of  the  shaft  and 
its  attached  parts  serves  to  hold  the  rubbing  surfaces  in  contact, 
and  the  lost  motion  is  taken  up  by  the  shaft  following  down  as  wear 
occurs.  In  collar  thrust  bearings  for  horizontal  shafts  the  design 
is  such  that  the  bearing  for  each  collar  is  separate  and  adjustable. 
The  pressure  on  the  different  collars  may  thus  be  equalized.* 

109.  Lubrication  of  Journals.  —  The  best  method  of  lubrication  is 
that  in  which  the  rubbing  surfaces  are  constantly  submerged  in  a 
bath  of  lubricating  fluid.  This  method  should  be  employed  when- 
ever possible,  if  the  pressure  and  surface  velocity  are  high.  Unfor- 
tunately it  cannot  be  used  in  the  majority  of  cases.  Let/,  Fig. 
121,  represent  a  journal  with  its  box,  and  let  A^  B,  and  C  be  oil 
holes.  If  oil  be  introduced  into  the  hole  A,  it  will  tend  to  flow  out 
from  between  the  rubbing  surfaces  by  the  shortest  way  ;  i.  e.,  it  will 
come  out  at  D.  A  small  amount  will  probably  go  toward  the  other 
end  of  the  box  because  of  capillary  attraction,  but  usually  none  of 
it  will  reach  the  middle  of  the  box.  If  oil  be  introduced  at  C,  it 
will  come  out  at  E.  A  constant  feed,  therefore,  might  be  main- 
tained at  A  and  C,  and  yet  the  middle  of  the  box  might  run  dry. 
If  the  oil  be  introduced  at  B,  however,  it  tends  to  flow  equally  in 
both  directions,  and  the  entire  journal  is  lubricated.  The  con- 
clusion follows  that  oil  ought,  when  possible,  to  be  introduced  at 
the  middle  of  the  length  of  a  cylindrical  journal.  If  a  conical 
journal  runs  at  a  high  velocity,  the  oil  under  the  influence  of  cen- 
trifugal force  tends  to  go  to  the  large  end  of  the  cone,  and  therefore 
the  oil  should  be  introduced  at  the  small  end  to  insure  its  distribu- 
tion over  the  entire  journal  surface. 

If  the  end  of  a  vertical  thrust  journal,  whose  outline  is  a  cone  or 
tractrix,  as  in  Fig.  122,  dips  into  a  bath  of  oil,  B,  the  oil  will  be 

*  For  complete  and  varied  details  of  marine  thrust  bearings  see  **  Maw's 
Modern  Practice  in  Marine  Engineering." 


124  MACHINE    DESIGN. 

carried  by  its  centrifugal  force,  if  the  velocity  be  high,  up  between 
the  rubbing  surfaces,  and  will  be  delivered  into  the  groove  AA.  If 
holes  connect  A  and  B,  gravity  will  return  the  oil  to  B,  and  a  con- 
stant circulation  will  be  maintained.  If  the  thrust  journal  has 
simply  a  flat  end,  as  in  Fig.  128,  the  oil  should  be  supplied  at  the 
centre  of  the  bearing  ;  centrifugal  force  will  then  distribute  it  over 
the  entire  surface.  Vertical  shaft  thrust  journals  may  usually  be 
arranged  to  run  in  an  oil  bath.  Marine  collar  thrust  journals  are 
always  arranged  to  run  in  an  oil  bath. 

Sometimes  a  journal  is  stationary  and  the  box  rotates  about  it, 
as  in  the  case  of  a  loose  pulley,  Fig.  124.  If  the  oil  is  introduced 
into  a  tube  A,  as  is  often  done,  its  centrifugal  force  will  carry  it 
away  from  the  rubbing  surface.  But  if  a  hole  is  drilled  in  the  axis 
of  the  journal,  the  lubricant  introduced  into  it  will  be  carried  to  the 
rubbing  surfaces  as  required.  If  a  journal  is  carried  in  a  rotating 
part  at  a  considerable  distance  from  the  axis  of  rotation,  and  it 
requires  to  be  oiled  while  in  motion,  a  channel  may  be  provided 
from  the  axis  of  rotation  where  oil  may  be  introduced  conveniently, 
to  the  rubbing  surfaces,  and  the  oil  will  be  carried  out  by  centrifu- 
gal force.  Thus  Fig.  125  shows  an  engine  crank  in  section.  Oil 
is  introduced  at  0,  and  centrifugal  force  carries  it  through  the 
channel  provided  to  a,  where  it  serves  to  lubricate  the  rubbing  sur- 
faces of  the  crank-pin  and  its  box.  If  a  journal  is  carried  in  a 
reciprocating  machine  part,  and  requires  to  be  oiled  while  in 
motion,  the  ''wick  and  wiper"  method  is  one  of  the  best.  See  Fig. 
126.  An  ordinary  oil  cup  with  an  adjustable  feed  is  mounted  in  a 
proper  position  opposite  the  end  of  the  stroke  of  the  reciprocating 
part,  and  a  piece  of  flat  wick  projects  from  its  delivery  tube.  A 
drop  of  oil  runs  down  and  hangs  suspended  at  its  end.  Another 
oil  cup  is  attached  to  the  reciprocpting  part,  which  carries  a  hooked 
"  wiper,"  B.  The  delivery  tube  from  C  leads  to  the  rubbing  surfaces 
to  be  lubricated.  When  the  reciprocating  part  reaches  the  end  of 
its  stroke  the  wiper  picks  off  the  drop  of  oil  from  the  wick,  and  it 
runs  down  into  the  oil  cup  C,  and  thence  to  the  surfaces  to  be 
lubricated.     This  method  applies  to  the  oiling  of  the  cross-head  pin 


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[Xj'iriVBBSITT] 


DESIGN    OF    JOURNALS.  125 

of  a  steam  engine.  The  same  method  is  sometimes  applied  to  the 
crank-pin,  but  here,  through  a  part  of  the  revolution,  the  tendency 
of  the  centrifugal  force  is  to  force  the  oil  out  of  the  cup,  and  there- 
fore the  plan  of  oiling  from  the  axis  is  probably  preferable. 

When  journals  are  lubricated  by  feed  oilers,  and  are  so  located 
as  not  to  attract  attention  if  the  lubrication  should  fail  for  any 
reason,  "tallow  boxes"  are  used.  These  are  cup-like  depressions 
usually  cast  in  the  box  cap,  and  communicating  by  means  of  an 
oil  hole  with  the  rubbing  surface.  These  cups  are  filled  with  grease 
that  is  solid  at  the  ordinary  temperature  of  the  box,  but  if  there 
is  the  least  rise  of  temperature  because  of  the  failure  of  the  oil 
supply,  the  grease  melts  and  runs  to  the  rubbing  surfaces,  and  sup- 
plies the  lubrication  temporarily.  This  safety  device  is  used  very 
commonly  on  line  shaft  journals. 

The  most  common  forms  of  feed  oilers  are  :  I.  The  oil  cup  with 
an  adjustable  valve  that  controls  the  rate  of  flow.  II.  The  oil 
cup  with  a  wick  feed.  Fig.  127.  The  delivery  has  a  tube  in- 
serted in  it  which  projects  nearly  to  the  top  of  the  cup.  In  this 
tube  a  piece  of  wicking  is  inserted,  and  its  end  dips  into  the  oil  in 
the  cup.  The  wick,  by  capillary  attraction,  carries  the  oil  slowly 
and  continuously  over  through  the  tube  to  the  rubbing  surfaces. 
III.  The  cup  with  a  copper  rod.  Fig.  128.  The  oil  cup  is  filled 
with  grease  that  melts  with  a  very  slight  elevation  of  temperature, 
and  ^  is  a  small  copper  rod  dropped  into  the  delivery  tube  and 
resting  on  the  surface  of  the  journal.  The  slight  friction  between 
the  rod  and  the  journal  warms  the  rod  and  it  melts  the  grease 
in  contact  with  it,  which  runs  down  the  rod  to  the  rubbing  sur- 
face. IV.  Sometimes  a  part  of  the  surface  of  the  bottom  half  of 
the  box  is  cut  away  and  a  felt  pad  is  inserted,  its  bottom  being  in 
contact  with  an  oil  bath.  This  pad  rubs  against  the  surface  of  the 
journal,  is  kept  constantly  soaked  with  oil,  and  maintains  lubrication. 


chaptp:r  XL 


BLIDING     SURFACES. 


110.  So  much  of  the  accuracy  of  action  of  machines  depends  on 
the  sliding  surfaces  that  their  design  deserves  the  most  careful 
attention.  The  perfection  of  the  cross-sectional  outline  of  the  cylin- 
drical or  conical  forms  produced  in  the  lathe,  depends  on  the  per- 
fection of  form  of  the  spindle.  But  the  perfection  of  the  outlines  of 
a  section  through  the  axis  depends  on  the  accuracy  of  the  sliding 
surfaces.  All  of  the  surfaces  produced  by  planers,  and  most  of 
those  produced  by  milling  machines,  are  dependent  for  accuracy  on 
the  sliding  surfaces  in  the  machine. 

Suppose  that  the  short  block  A,  Fig.  129,  is  the  slider  of  a  slider- 
crank  chain,  and  that  it  slides  on  a  relatively  long  guide,  D.  The 
direction  of  rotation  of  the  crank,  a,  is  as  indicated  by  the  arrow. 
B  and  C  are  the  extreme  positions  of  the  slider.  The  pressure 
between  the  slider  and  the  guide  is  greatest  at  the  mid-position,  A, 
and  at  the  extreme  positions,  B  and  C,  it  is  only  the  pressure  due 
to  the  weight  of  the  slider.  Also  the  velocity  is  a  maximum  when 
the  slider  is  in  its  mid-position,  and  decreases  toward  the  ends, 
becoming  zero  when  the  crank  a  is  on  its  centre.  The  work  of  fric- 
tion is  therefore  greatest  at  the  middle,  and  is  very  small  near  the 
ends.  Therefore  the  wear  would  be  greatest  at  the  middle,  and  the 
guide  would  wear  concave.  If  now  the  accuracy  of  a  machine's 
working  depends  on  the  perfection  of  vl's  rectilinear  motion,  that 
accuracy  will  be  destroyed  as  the  guide  I)  wears.  Suppose  a  gib, 
EFG,  to  be  attached  to  A,  Fig.  130,  and  to  engage  with  Z>,  as  shown, 
to  prevent  vertical  looseness  between  A  and  D.  If  this  gib  be  taken 
up  to  compensate  wear  after  it  has  occurred,  it  will  be  loose  in  the 


SLIDING    SURFACES.  127 

middle  position  when  it  is  tight  at  the  ends,  because  of  the  unequal 
wear.  Suppose  that  A  and  D  are  made  of  equal  length,  as  in  Fig. 
131.  Then  when  A  is  in  the  mid-position  corresponding  to  maxi- 
mum pressure,  velocity,  and  wear,  it  is  in  contact  with  D  through- 
out its  entire  surface,  and  the  wear  is  therefore  the  same  in  all  parts 
of  the  surface.  The  slider  retains  its  accuracy  of  rectilinear  motion 
regardless  of  the  amount  of  wear,  the  gib  may  be  set  up,  and  will 
be  equally  tight  in  all  positions. 

If  A  and  B,  Fig.  132,  are  the  extreme  positions  of  a  slider,  /) 
being  the  guide,  a  shoulder  would  be  finally  worn  at  C.  It  would 
be  better  to  cut  away  the  material  of  the  guide,  as  shown  by  the 
dotted  line.  Slides  should  always  "wipe  over"  the  ends  of  the 
guide  when  it  is  possible.  Sometimes  it  is  necessary  to  vary  the 
length  of  stroke  of  a  slider,  and  also  to  change  its  position  relatively 
to  the  guide.  Examples  :  "  Cutter  bars  "  of  slotting  and  shaping 
machines.  In  some  of  these  positions,  therefore,  there  will  be  a 
tendency  to  wear  shoulders  in  the  guide  and  also  in  the  cutter  bar 
itself.  This  difficulty  is  overcome  if  the  slide  and  guide  are  made 
of  equal  length,  and  the  design  is  such  that  when  it  is  necessary  to 
change  the  position  of  the  cutter  bar  that  is  attached  to  the  slide, 
the  position  of  the  guide  may  be  also  changed  so  that  the  relative 
position  of  slide  and  guide  remains  the  same.  The  slider  surface 
will  then  just  completely  cover  the  surface  of  the  guide  in  the  mid- 
position,  and  the  slider  will  wipe  over  each  end  of  the  guide,  what- 
ever the  length  of  the  stroke. 

In  many  cases  it  is  impossible  to  make  the  slider  and  guide  of 
equal  length.  Thus  a  lathe  carriage  cannot  be  as  long  as  the  bed  ; 
a  planer  table  cannot  be  as  long  as  the  planer  bed,  nor  a  planer 
saddle  as  long  as  the  cross-head.  When  these  conditions  exist 
especial  care  should  be  given  to  the  following  :  I.  The  bearing 
surface  should  be  made  so  large  in  proportion  to  the  pressure  to  be 
sustained  that  the  maintenance  of  lubrication  shall  be  insured 
under  all  conditions.  11.  The  parts  which  carry  the  wearing  sur- 
faces should  be  made  so  rigid  that  there  shall  be  no  possibility  of 
the  localization  of  pressure  from  yielding. 


128  MACHINE    DESIGN. 

111.  As  to  form,  guides  may  be  divided  into  two  classes  :  angu- 
lar guides  and  flat  guides.  Fig.  133  a,  shows  an  angular  guide, 
the  pressure  being  applied  as  shown.  The  advantage  of  this  form 
is,  that  as  the  rubbing  surfaces  wear,  the  slide  follows  down  and 
takes  up  both  the  vertical  and  lateral  wear.  The  objection  to 
this  form  is  that  the  pressure  is  not  applied  at  right  angles  to 
the  wearing  surfaces,  as  it  is  in  the  flat  guide  shown  in  b.  But 
in  6  a  gib,  A,  must  be  provided  to  take  up  the  lateral  wear.  The 
gib  is  either  a  wedge  or  a  strip  with  parallel  sides  backed  up  by 
screws.  Guides  of  these  forms  are  used  for  planer  tables.  The 
weight  of  the  table  itself  holds  the  surfaces  in  contact,  and  if  the 
table  is  light  the  tendency  of  a  heavy  side  cut  would  be  to  force  the 
table  up  one  of  the  angular  surfaces  away  from  the  other.  If  the 
table  is  very  heavy,  however,  there  is  little  danger  of  this,  and 
hence  the  angular  guides  of  large  planers  are  much  flatter  than 
those  of  smaller  ones.  In  some  cases  one  of  the  guides  of  a  planer 
table  is  angular  and  the  other  is  flat.  The  side  bearings  of  the  flat 
guide  may  then  be  omitted,  as  the  lateral  wear  is  taken  up  by  the 
angular  guide.  This  arrangement  is  undoubtedly  good  if  both 
guides  wear  down  equally  fast. 

112.  Fig.  134  shows  three  forms  of  sliding  surfaces  such  as  are 
used  for  the  cross  slide  of  lathes,  the  vertical  slide  of  shapers,  the 
table  slide  of  milling  machines,  etc.  ^  is  a  taper  gib  that  is  forced 
in  by  a  screw  at  D  to  take  up  wear.  When  it  is  necessary  to  take 
up  wear  at  B,  the  screw  may  be  loosened  and  a  shim  or  liner  may 
be  inserted  between  the  surfaces  at  a.  0  is  a  thin  gib,  and  the  wear 
is  taken  up  by  means  of  several  screws  like  the  one  shown.  This 
form  is  not  so  satisfactory  as  the  wedge  gib,  as  the  bearing  is  chiefly 
under  the  points  of  the  screws,  the  gib  being  thin  and  yielding, 
whereas  in  the  wedge  there  is  complete  contact  between  the  metallic 
surfaces. 

113.  The  sliding  surfaces  thus  far  considered  have  to  be  designed 
80  that  there  will  be  no  lost  motion  while  they  are  moving,  because 
they  are  required  to  move  while  the  machine  is  in  operation.  The 
gibs  have  to  be  carefully  designed  and  accurately  set  so  that  the 


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SLIDING   SURFACES.  129 

moving  part  shall  be  just  "  tight  and  looee,"  i.  e.,  so  that  it  shall  be 
free  to  move,  without  lost  motion  to  interfere  with  the  accurate 
action  of  the  machine.  There  is,  however,  another  class  of  sliding 
parts,  like  the  sliding  head  of  a  drill  press,  or  the  tail  stock  of  a 
lathe,  that  are  never  required  to  move  while  the  machine  is  in 
operation.  It  is  only  required  that  they  shall  be  capable  of  being 
fastened  accurately  in  a  required  position,  their  movement  being 
simply  to  readjust  them  to  other  conditions  of  work,  while  the 
machine  is  at  rest.  No  gib  is  necessary  and  no  accuracy  of  motion 
is  required.  It  is  simply  necessary  to  insure  that  their  position  is 
accurate  when  they  are  clamped  for  the  special  work  to  be  done. 


CHAPTER   XII. 

BOLTS   AND   SCREWS    AS    MACHINE    FASTENINGS. 

114.  Classification  may  be  made  as  follows :  I.  Bolts.  II. 
Studs.  III.  Cap  Screws,  or  Tap  Bolts.  IV.  Set  Screws.  V.  Ma- 
chine Screws. 

A  "bolt"  consists  of  a  head  and  round  body  on  which  a  thread 
is  cut,  and  upon  which  a  nut  is  screwed.  When  a  bolt  is  used  to 
connect  machine  parts,  a  hole  the  size  of  the  body  of  the  bolt  is 
drilled  entirely  through  both  parts,  the  bolt  is  put  through,  and  the 
nut  screwed  down  upon  the  washer.     See  Fig.  135. 

A  "  stud  "  is  a  piece  of  round  metal  with  a  thread  cut  upon  each 
end.  One  end  is  screwed  into  a  tapped  hole  in  some  part  of  a 
machine,  and  the  piece  to  be  held  against  it,  having  a  hole  the  size 
of  the  body  of  the  stud,  is  put  on,  and  a  nut  is  screwed  upon  the 
other  end  of  the  stud  against  the  piece  to  be  held.     See  Fig.  136. 

A  "cap  screw"  is  a  substitute  for  a  stud,  and  consists  of  a  head 
and  body  on  which  a  thread  is  cut.  See  Fig.  137.  The  screw  is 
passed  through  the  removable  part  and  screwed  into  a  tapped  hole 
in  the  part  to  which  it  is  attached.  A  cap  screw  is  a  stud  with  a 
head  substituted  for  the  nut. 

A  hole  should  never  be  tapped  into  a  cast  iron  machine  part 
when  it  can  be  avoided.  Cast  iron  is  not  good  material  for  the 
thread  of  a  nut,  since  it  is  weak  and  brittle  and  tends  to  crumble. 
In  very  many  cases,  however,  it  is  absolutely  necessary  to  tap  into 
cast  iron.  It  is  then  better  to  use  studs,  if  the  attached  part  needs  to 
be  removed  often,  because  studs  are  put  in  once  for  all,  and  the  cast 
iron  thread  would  be  worn  out  eventually  if  cap  screws  were  used. 

When  one  machine  part  surrounds  another,  as  a  pulley  hub 


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UITIVERSITT] 


BOLTS    AND    SCREWS    AS    MACHINE    FASTENINGS.  131 

surrounds  a  shaft,  relative  motion  of  the  two  is  often  prevented  by- 
means  of  a  "set  screw  "which  is  a  threaded  body  with  a  small 
square  head.  Fig.  138.  The  end  is  either  pointed  as  in  Fig.  138  6, 
or  cupped  as  in  c,  and  is  forced  against  the  inner  part  by  screwing 
through  a  tapped  hole  in  the  outer  part. 

The  term  "  machine  screws"  covers  many  forms  of  small  screws, 
usually  with  screw-driver  heads.  All  of  the  kinds  given  in  this 
classification  are  made  in  great  variety  of  size,  form,  length,  etc. 

115.  Design  of  Bolts  and  Screws.  —  Tensile  stress  is  induced  in  a 
bolt  by  tightening  it.  This  stress  may  equal,  or  very  greatly  exceed, 
the  tensile  stress  due  to  working  forces.  The  stress  due  to  tightening 
may  be  approximately  found  as  follows  :  In  Fig.  139,  a  force  P  is 
applied  to  the  wrench  handle  at  A.  During  the  turning  of  the 
wrench  the  ratio  of  movement  of  the  point  A  to  the  movement  of 
the  nut  axially  =  2-rd  to  p,  in  which  i^  is  the  wrench  lever  arm,  and 
p  is  the  pitch  of  the  screw  (axial  distance  between  threads).  If 
there  were  no  frictional  resistance,  the  force  P  and  the  resistance  to 
tension  of  the  bolt,  ■==  T,  would  be  in  equilibrium  at  the  instant  of 
tightening  up,  and  the  following  equation  would  be  true: 

P2r:l=Tp. 

But  force  P  must  also  overcome  the  frictional  resistance  between 
the  nut  and  the  screw  thread,  and  between  the  nut  and  washer. 
This  resistance  R=^Tf;  in  which  /=:the  coefficient  of  friction. 
The  radius,  r,  of  this  resistance,  R,  may  be  assumed,  for  this  ap- 
proximation, equal  to  the  radius  of  the  top  of  the  screw  threads  X 
1'5.  The  ratio  of  movement  of  R  to  axial  movement  of  the  nut  ^= 
2-r  to  p.  At  the  instant  of  tightening  up  there  is  equilibrium 
between  P,  R,  and  T,  and  the  following  equation  is  true: 

27:pl=Tp-^  27rRr  ;  substituting  for  R^JT, 

=  Tp-i-  2r.Tfr, 

2nPl 
whence  T  =  — ttttf  • 

p  -f  2T.fr 


1B2  MACHINE    DESIGN. 

For  a  Y  ^o\i  the  values  are  I  =^  8",  p  ■=  0'077",/=  0-15,  r  =  0'375. 
Making  P  ---  1  lb.,  T'=  116  lbs.  Hence,  for  every  pound  applied  at 
A  there  results  116  lbs.  tensile  stress  in  the  bolt.  The  ultimate 
strength  of  the  bolt  -4-116  equals  the  force  applied  at  A  necessary 
to  break  the  bolt.  The  area  of  cross-section  at  the  bottom  of  the 
thread  of  a  i"  bolt  =  0'12  sq.  in.  Assume  the  ultimate  strength  of 
the  material  of  the  bolt  =  50000  lbs.  per  sq.  in.  The  ultimate  ten- 
sile strength  of  the  bolt  -=  50000  X  012  =  6000  lbs.  Then  the  force 
at  A  to  break  the  bolt  =  6000  -^  116  =  52  lbs.  nearly.  This  is  prob- 
ably not  the  actual  force  that  would  break  the  bolt,  because  the 
assumptions  are  probably  somewhat  inaccurate  ;  but  it  indicates 
that  a  half  inch  bolt  may  be  pulled  in  two  by  force  applied  by  a 
man  to  the  wrench  handle.  For  a  |"  bolt  the  force  becomes  about 
100  lbs. 

Suppose  a  nut  screwed  up  with  a  resulting  tensile  stress  in  the 
bolt  =  p.  Suppose  that  a  gradually  increasing  working  force,  =  p^, 
is  applied.  If  there  were  no  elongation  of  the  bolt,  the  total  stress 
in  the  bolt  would  equal  p  -)-  py  But  elongation  does  result  from 
the  application  of  p^,  and  ^  is  reduced,  and  the  total  stress  in  the 
bolt  is  less  than  p  -f-  p^ 

116.  Illustration.  —  In  Fig.  140  the  tensile  stress  in  the  bolt  due 
to  screwing  up  =p.  The  pressure  between  the  surfaces  in  contact 
at  CD  is  therefore  =^  p.  Suppose  a  working  force,  j9„  applied  tend- 
ing to  separate  A  and  B.  The  bolt  yields  to  the  increased  stress, 
and  the  pressure  at  CD  is  reduced.  The  tensile  stress  in  the  bolt  is 
now  equal  to  the  working  force  plus  the  reduced  pressure  at  CD. 
When  the  working  force  reduces  the  pressure  at  CD  to  zero,  the 
stress  in  the  bolt  =  the  working  force,  and  if  CD  were  a  steam 
joint,  it  would  leak. 

117.  It  is  required  to  design  the  fastenings  to  hold  on  the  steam 
chest  cover  of  a  steam  engine.  The  opening  to  be  covered  is  rec- 
tangular, 10"x  12".  The  maximum  steam  pressure  is  100  lbs.  per 
square  inch.  The  joint  must  be  held  steam  tight.  Short  unyield- 
ing fastenings  are  therefore  best  suited  to  the  purpose,  and  studs 
will  be  used.     They  will  be  made  of  machinery  steel  of  60000  lbs. 


BOLTS   AND    SCREWS    AS    MACHINE    FASTENINGS.  133 

tensile  strength,  and  will  be  |"  outside  diameter  because  smaller 
studs  may  be  ruptured  by  the  force  applied  in  screwing  up.  It  will 
be  assumed  that  the  stress  on  the  studs  is  equal  to  p  -\-  p^,  i.  e.,  the 
stress  due  to  screwing  up,  plus  the  working  stress.  This  assump- 
tion cannot  be  exactly  true,  as  is  seen  from  the  preceding  illustra- 
tion ;  but  the  resulting  error  is  on  the  safe  side.  The  diameter  of  a 
I"  stud  at  the  bottom  of  the  thread  is  0'62",  and  the  area  is  = 
0*62^  X  7^  -^  4  =  0*3  sq.  in.  The  ultimate  strength  of  the  stud  is, 
therefore,  0*3  X  60000  =  18000.  The  factor  of  safety  may  be  4 
because  the  stress  member  is  of  resilient  material  and  is  not  subject 
to  shocks.  Then  the  allowable  stress  on  each  stud  would  be  equal 
to  18000  -^-  4  =  4500.  The  stress  due  to  screwing  up,  subtracted 
from  the  total  allowable  stress  gives  the  allowable  working  stress  in 
the  stud.  The  stress  due  to  screwing  up  may  be  found  from  the 
above  equation  for  T.  Assuming  P=:  30  lbs.;  /  =  10";  /==015; 
r  :=  056;  gives  7"=  3000  lbs.  nearly.  The  allowable  working  stress 
in  each  stud  equals  4500  —  3000  =  1500  lbs.  The  maximum  work- 
ing force  on  the  cover  equals  the  area  of  the  opening  covered,  multi- 
plied by  the  maximum  working  pressure  per  square  inch  ;  = 
10  X  12  =-  120  sq.  in.  X  100  lbs.  per  sq.  in.  =-  12000  lbs.  This  di- 
vided by  the  allowable  working  pressure  for  each  stud  gives  the 
number  of  studs  required  for  strength,  -=12000  -v-  1500  =  8.  There- 
fore 8  studs  will  serve  for  strength.  But  in  order  to  make  a  steam 
tight  joint,  with  a  reasonable  thickness  of  steam  chest  cover,  the 
distance  between  the  stud  centres  should  not  be  greater  than  about 
A\".  The  opening  is  10"x  12",  as  shown  in  Fig.  141.  There  must  be 
a  band  about  |"  wide  around  this  for  making  the  joint  upon  which 
the  studs  must  not  encroach.  This  makes  the  distance  between 
the  vertical  rows  of  studs  14",  and  between  the  horizontal  rows  12". 
The  whole  length  over  which  the  studs  are  to  be  distributed  then  =^ 
12  -f  12  +  14  +  14  =  52".  If  they  are  4-5"  apart  the  number  of 
studs  =  52  -V-  4*5  --=  11 '5.  Hence  it  is  necessary  to  use  12  studs  to 
make  the  joint  tight,  while  8  would  serve  for  strength.  In  order 
to  get  a  symmetrical  arrangement,  it  will  probably  be  necessary  to 
use  14  studs.     The  number  of  studs  is  therefore  dictated   by  the 


134  MACHINE    DESIGN. 

conditions  necessary  to  maintain  a  tight  steam  joint,  and  not  by 
the  applied  forces. 

118.  The  elongation  of  a  bolt  with  a  given  total  stress,  depends 
upon  the  length  and  area  of  its  least  cross-section.  Suppose,  to 
illustrate,  that  the  bolt.  Fig.  142,  has  a  reduced  section  over  a 
length  I  as  shown.  This  portion  A  has  less  cross-sectional  area 
than  the  rest  of  the  bolt,  and  when  any  tensile  force  is  applied,  the 
resulting  unit  stress  will  be  greater  in  A  than  elsewhere..  The  unit 
strain,  or  elongation,  will  be  proportionately  greater,  up  to  the 
elastic  limit ;  and  if  the  elastic  limit  is  exceeded  in  the  portion  A, 
the  elongation  there  will  be  far  greater  than  elsewhere.  If  there  is 
much  difference  of  area  and  the  bolt  is  tested  to  rupture,  the  elonga- 
tion will  be  chiefly  at  A.  There  would  be.  a  certain  elongation  per 
inch  of  A  at  rupture.  Hence,  the  greater  the  length  of  A,  the 
greater  the  total  elongation  of  the  bolt.  If  the  bolt  had  not  been 
reduced  at  A,  the  minimum  section  would  be  at  the  root  of  the 
screw  threads.  The  axial  length  of  this  section  is  very  small. 
Hence  the  elongation  at  rupture  would  be  small.  Suppose  there 
are  two  bolts,  A  with,  and  B  without,  the  reduced  section.  They 
are  alike  in  other  respects,  They,  are  subjected  to  equal  tensile 
shocks.  Let  the  energy  of  the  shock  =  E.  This  energy  is  divided 
into  force  and  space  factors  by  the  resistance  of  the  bolts.  The 
space  factor  equals  the  elongation  of  the  bolt.  This  is  greater 
in  A  than  in  B  because  of  the  yielding  of  the  reduced  section. 
But  the  product  of  force  and  space  factors  is  the  same  in  both 
bolts,  =  J5^ ;  hence  the  resulting  stress  in  the  minimum  section 
is  less  for  A  than  for  B.  The  stress  in  A  may  be  less  than  the 
breaking  stress ;  while  the  greater  stress  in  B  may  break  it. 
The  capacity  of  the  bolt  to  resist  shock  is  therefore  increased  hy 
lengthening  its  minimum  section  to  increase  the  yielding  and  reduce 
stress.  This  is  not  only  true  of  bolts,  but  of  all  stress  members  in 
machines. 

The  whole  body  of  the  bolt  might  have  been  reduced,  as  shown 
by  the  dotted  lines  in  Fig.  142,  with  resulting  increase  of  capacity 
to  resist   shock.     Turning   down    a    bolt,  however,  weakens   it   to 


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UiriVBRSITT] 


BOLTS    AND   SCREWS    AS    MACHINE    FASTENINGS.  135 

resist  torsion  and  flexure,  because  it  takes  off  the  material  which 
is  most  effective  in  producing  large  polar  and  rectangular  moments 
of  inertia  of  cross-section.  If  the  cross-sectional  area  is  reduced  b}'' 
drilling  a  hole  as  shown  in  Fig.  143,  the  torsional  and  transverse 
strength  is  but  slightly  decreased,  but  the  elongation  will  be  as 
great,  with  the  same  area,  as  if  the  area  had  been  reduced  by  turn- 
ing down. 

119.  Prof.  Sweet  had  a  set  of  bolts  prepared  for  special  test. 
The  bolts  were  1^"  diameter  and  about  12"  long.  They  were  made 
of  high  grade  wrought  iron,  and  were  duplicates  of  the  bolts  used  at 
the  crank  end  of  the  connecting-rods  of  one  of  the  standard  sizes  of 
the  Straight  Line  Engine.  Half  of  the  bolts  were  left  solid,  while 
the  other  half  were  carefully  drilled  to  give  them  uniform  cross- 
sectional  area  throughout.  The  tests  were  made  under  the  direction 
of  Prof.  Carpenter  at  the  Sibley  College  Laboratory.  One  pair  of 
bolts  was  tested  to  rupture  by  tensile  force  gradually  applied.  The 
undrilled  bolt  broke  in  the  thread  with  a  total  elongation  of  0"25". 
The  drilled  bolt  broke  between  the  thread  and  the  bolt  head  with  a 
total  elongation  of  2*25".  If  it  be  assumed  that  the  mean  force  ap- 
plied was  the  same  in  both  cases,  it  follows  that  the  total  resilience 
of  the  drilled  bolt  was  nine  times  as  great  as  that  of  the  solid  one. 
"Drop  tests,"  i.  e.,  tests  which  brought  tensile  shock  to  bear  upon 
the  bolts,  were  made  on  other  similar  pairs  of  bolts,  which  tended  to 
confirm  the  general  conclusion. 

120.  It  is  required  to  design  proper  fastenings  for  holding  on 
the  cap  of  a  connecting-rod  like  that  shown  in  Fig.  144.  These 
fastenings  are  required  to  sustain  shocks,  and  may  be  subjected  to 
a  maximum  accidental  stress  of  20000  lbs.  There  are  two  fasten- 
ings, and  therefore  each  must  be  capable  of  sustaining  safely  a 
stress  of  10000  lbs.  They  should  be  designed  to  yield  as  much  as 
is  consistent  with  strength  ;  in  other  words,  they  should  be  tensile 
springs  to  cushion  shocks  and  thereby  reduce  the  resulting  force 
they  have  to  sustain.  Bolts  should  therefore  be  used,  and  the 
weakest  section  should  be  made  as  long  as  possible.  Wrought  iron 
will  be  used  whose  tensile  strength  is  50000  lbs.  per  square  inch. 


136  MACHINE    DESIGN. 

The  stress  given  is  the  maximum  accidental  stress,  and  is  four 
times  the  working  stress.  It  is  not,  therefore,  necessary  to  give  the 
bolts  great  excess  of  strength  over  that  necessary  to  resist  actual 
rupture  by  the  accidental  force.  Let  the  factor  of  safety  be  2. 
Then  the  cross-sectional  area  of  each  bolt  must  be  such  that  it  will 
just  sustain  10000  X  2  =  20000  lbs.  This  area  =  20000  -^-  50000 
=  0"4  square  inches.  This  area  corresponds  to  a  diameter  of  0*71", 
and  that  is  nearly  the  diameter  of  a  |"  bolt  at  the  bottom  of  the 
thread  ;  hence  ^"  bolts  will  be  used.  The  cross-sectional  area  of 
the  body  of  the  bolt  must  now  be  made  at  least  as  small  as  that  at 
the  bottom  of  the  thread.     This  may  be  accomplished  by  drilling. 

121.  When  bolts  are  subjected  to  constant  vibration  there  is  a 
tendency  for  the  nuts  to  loosen.  There  are  many  ways  to  prevent 
this,  but  the  most  common  one  is  by  the  use  of  jamb  nuts.  Two 
nuts  are  screwed  on  the  bolt ;  the  under  one  is  set  up  against  the 
surface  of  the  part  to  be  held  in  place,  and  then  while  this  nut  is 
held  with  a  wrench  the  other  nut  is  screwed  up  against  it  tightly. 
Suppose  that  the  bolt  has  its  axis  vertical  and  that  the  nuts  are 
screwed  on  the  upper  end.  The  nuts  being  screwed  against  each 
other  the  upper  one  has  its  internal  screw  surfaces  forced  against 
the  under  screw  surfaces  of  the  bolt,  and  if  there  is  any  lost  motion, 
as  there  almost  always  is,  there  will  be  no  contact  between  the 
upper  surfaces  of  the  screw  on  the  bolt  and  the  threads  of  the  nut. 
Just  the  reverse  is  true  of  the  under  nut ;  i.  e.,  there  is  no  contact 
between  the  under  surfaces  of  the  threads  on  the  bolt  and  the 
'threads  on  the  nut.  Therefore  no  pressure  that  comes  from  the 
under  side  of  the  under  nut  can  be  communicated  to  the  bolt 
through  the  under  nut  directly,  but  it  must  be  received  by  the 
upper  nut  and  communicated  by  it  to  the  bolt,  since  it  is  the  upper 
nut  alone  that  has  contact  with  the  under  surfaces  of  the  thread. 
Therefore  the  jamb  nut,  which  is  usually  made  about  half  as  thick 
as  the  other,  should  always  be  put  on  next  to  the  surface  of  the 
piece  to  be  held  in  place. 


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CHAPTER  XIII. 

MEANS    FOR    PREVENTING    RELATIVE    ROTATION. 

122.  Keys  are  chiefly  used  to  prevent  relative  rotation  between 
shafts  and  the  pulleys,  gears,  etc.  which  they  support.  Keys  may 
be  divided  into  parallel  keys,  taper  keys,  and  feathers  or  splines. 

For  a  parallel  key  the ''  seat,"  both  in  the  shaft  and  the  attached 
part,  has  parallel  sides,  and  the  key  simply  prevents  relative  ro- 
tary motion.  Motion  parallel  to  the  axis  of  the  shaft  must  be  pre- 
vented by  some  other  means  ;  as  by  set  screws  which  bear  upon  the 
top  surface  of  the  key,  as  shown  in  Fig.  145.  A  parallel  key 
should  fit  accurately  on  the  sides  and  loosely  at  the  top  and  bottom. 

A  taper  key  has  parallel  sides  and  has  its  top  and  bottom  sur- 
faces tapered,  and  is  made  to  fit  on  all  four  surfaces,  being  driven 
tightly  "  home."  It  prevents  relative  motion  of  any  kind  between 
the  parts  connected.  If  a  key  of  this  kind  has  a  head,  as  shown  in 
Fig.  146,  it  is  called  a  "  draw  key,"  because  it  is  drawn  out  when 
necessary,  by  driving  a  wedge  between  the  hub  of  the  attached  part 
and  the  head  of  the  key.  When  a  taper  key  has  no  head  it  is  re- 
moved by  driving  against  the  point  with  a  "  key  drift." 

Feathers  or  Splines  are  keys  that  prevent  relative  rotation,  but 
purposely  allow  axial  motion.  They  are  sometimes  made  fast  in 
the  shaft,  as  in  Fig.  147,  and  there  is  a  key  "  way  "  in  the  attached 
part  that  slides  along  the  shaft.  Sometimes  the  feather  is  fast- 
ened in  the  hub  of  the  attached  part,  as  shown  in  Fig.  148,  and 
slides  in  a  long  key  way  in  the  shaft. 

John  Richards'  rule  for  keys  is  (see  Fig.  149)  w  =  j.     t  has 

such  value  that  «  =  30°.      This  rule  is  deviated  from  somewhat,  as 

i8 


188  MACHINE    DESIGN. 

shown  by  the  following  table  taken   from   Richards'   "  Manual  of 
Machine  Construction,"  page  58. 


w=l     li 

H 

If    2 

2i 

3 

3^ 

4 

5 

6 

7 

8 

d  =  i      5,6 

i 

'■6       i 

1 

1 

1 

1 

li 

1| 

U 

IJ 

i  =  ha     \6 

i 

'3.     5.6 

1 

'.6 

i 

s 

">6 

"^6 

1 

1 

When  d  exceeds  8"  two  or  more  keys  should  be  used,  and  w  may 
then  =^  d  -^-  16  ;  t  being  as  before  of  such  value  that  a  shall  =  30°. 
The  following  table  for  dimensions  for  parallel  keys  is  also  from 
Richards'  "Manual": 


d=l       li     li     1|      2      2^      3      3i      4 


W  =  h2       h-       ^2       "3^      ^^3,      ^^3^      ^V       9 


Also  this  for  feathers  : 

d  =  U     li     1|      2      2i     2i      3      3i      4     4i 


W  =  i  i         \6         ^6        -t  I         i 


16 


<=l  I        '.6        '.6        ^  i  I  «         I         J 

For  keying  hand  wheels  and  other  parts  that  are  not  subjected 
to  very  great  stress,  a  cheap  and  satisfactory  method  is  to  use  a 
round  key  driven  into  a  hole  drilled  in  the  joint,  as  in  Fig.  150.  If 
the  two  parts  are  of  different  material,  one  much  harder  than  the 
other,  this  method  should  not  be  used,  as  it  is  almost  impossible  in 
such  case  to  make  the  drill  follow  the  joint. 

The  taper  of  keys  varies  from  -J"  to  i"  to  the  foot. 

A  cotter  is  a  key  that  is  used  to  attach  parts  subjected  to  a  force 
of  tension  tending  to  separate  them.  Thus  piston  rods  are  often 
connected  to  both  piston  and  cross-head  in  this  way.  Also  the  sec- 
tions of  long  pump-rods,  etc. 

Fig.  151  shows  machine  parts  held  against  tension  by  cotters. 
It  is  seen  that  the  joint  may  yield  by  shearing  the  cotter  Sit  AB 


MEANS    FOR   PREVENTING    RELATIVE    ROTATION. 


139 


and  CD ;  or  by  shearing  CPQ  and  ARS  ;  by  shearing  on  the  sur- 
faces MO  and  LN \  or  by  tensile  rupture  of  the  rod  on  a  horizontal 
section  at  LM.  All  of  these  sections  should  be  sufficiently  large  to 
resist  the  maximum  stress  safely.  The  difficulty  is  usually  to  get 
LM  strong  enough  in  tension  ;  but  this  may  usually  be  accom- 
plished by  making  the  rod  larger,  or  the  cotter  thinner  and  wider. 
It  is  found  that  taper  surfaces  if  they  be  smooth  and  somewhat 
oily  will  just  cease  to  stick  together  when  the  taper  equals  1'5"  per 
foot.  The  taper  of  the  rod  in  Fig.  151  should  be  about  this  value 
in  order  that  it  may  be  removed  conveniently  when  necessary. 

123.  Shrink  and  Force  Fits. —  Relative  rotation  between  machine 
parts  is  also  prevented  sometimes  by  means  of  shrink  and  force  fits. 
In  the  former  the  shaft  is  made  larger  than  the  hole  in  the  part  to 
be  held  upon  it,  and  the  metal  surrounding  the  hole  is  heated,  usu- 
ally to  low  redness,  and  because  of  the  expansion  it  may  be  put  on 
the  shaft  and  on  cooling,  it  shrinks  and  "grips"  the  shaft.  A  key 
is  sometimes  used  in  addition  to  this. 

Force  fits  are  made  in  the  same  way  except  that  they  are  put 
together  cold,  either  by  driving  together  with  a  heavy  sledge  or  by 
forcing  together  by  hydraulic  pressure.  The  necessary  allowance 
for  forcing,  i.  e.,  the  excess  of  shaft  diameter  over  the  diameter  of 
the  hole,  is  given  in  the  following  table  : 


Inches. 


Diameter  of  Shaft 
Allowance  for  Forcing 


1 
0-004 


2 

0-005 


3 

0-006 


4 

0-006 


5 
0-007 


6 
0-008 


7 
0-008 


8 
0-009 


9 
0-01 


10 
0-01 


Experience  shows  that  with  this  allowance  a  steel  shaft  may  be 
forced  into  a  hole  in  cast  iron  bj'  a  total  pressure  of  from  40  to  90 
tons.  There  is  no  need  of  keying  when  parts  are  put  together  in 
this  way. 


CHAPTER  XIV. 

FORM    OF    MACHINE   PARTS   AS    DICTATED    BY   STRESS. 

124.  Suppose  that  A  and  B,  Fig.  152,  are  two  surfaces  in  a 
machine  to  be  joined  by  a  member  subjected  to  simple  tension. 
What  is  the  proper  form  for  the  member?  The  stress  in  all  sections 
of  the  member  at  right  angles  to  the  line  of  application,  AB,  of  the 
force,  will  be  the  same.  Therefore  the  areas  of  all  such  sections 
should  be  equal;  hence  the  outlines  of  the  member  should  be  straight 
lines  parallel  to  AB.  The  distance  of  the  material  from  the  axis 
AB  has  no  effect  on  its  ability  to  resist  tension.  Therefore  there  is 
nothing  in  the  character  of  the  stress  that  indicates  the  form  of  the 
cross-section  of  the  member.  The  form  most  cheaply  produced, 
both  in  the  rolling  mill  and  the  machine  shop,  is  the  cylindrical 
form.  Economy,  therefore,  dictates  the  circular  cross-section. 
After  the  required  area  necessary  for  safely  resisting  the  stress  is 
determined,  it  is  only  necessary  to  find  the  corresponding  diameter, 
and  it  will  be  the  diameter  of  all  sections  of  the  required  member 
if  they  are  made  circular.  Sometimes  in  order  to  get  a  more  har- 
monious design,  it  is  necessary  to  make  the  tension  member  just 
considered  of  rectangular  cross-section,  and  this  is  allowable  al- 
though it  almost  always  costs  more.  The  thin,  wide,  rectangular 
section  should  be  avoided,  however,  because  of  the  difficulty  of  in- 
suring a  uniform  distribution  of  stress.  A  unit  stress  might  result 
from  this  at  one  edge,  greater  than  the  strength  of  the  material, 
and  the  piece  would  yield  by  tearing,  although  the  average  stress 
might  not  have  exceeded  a  safe  value. 

If  the  stress  be  compression  instead  of  tension,  the  same  consid- 
erations dictate  its  form  as  long  as  it  is  a  "  short  block,"  i.  e.,  as  long 


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FORM    OF    MACHINE    PARTS   AS    DICTATED    BY    STRESS.  141 

as  the  ratio  of  length  to  lateral  dimensions  is  such  that  it  is  sure  to 
yield  by  crushing  instead  of  by  "  buckling."  A  short  block,  there- 
fore, should  have  its  longitudinal  outlines  parallel  to  its  axis,  and 
its  cross-section  may  be  of  any  form  that  economy  or  appearance 
may  dictate.  Care  should  be  taken,  however,  that  the  least  lateral 
dimension  of  the  member  be  not  made  so  small  that  it  is  thereby 
converted  into  a  "long  column." 

If  the  ratio  of  longitudinal  to  lateral  dimensions  is  such  that  the 
member  becomes  a  "long  column,"  the  conditions  that  dictate  the 
form  are  changed,  because  it  would  yield  by  buckling  or  flexure, 
instead  of  crushing.  The  strength  and  stiffness  of  a  long  column 
are  proportional  to  the  moment  of  inertia  of  the  cross-section  about 
a  gravity  axis  at  right  angles  to  the  plane  in  which  the  flexure  oc- 
curs. A  long  column  with  "fixed"  or  "rounded"  ends  has  a  tend- 
ency to  yield  by  buckling  which  is  equal  in  all  directions.  Therefore 
the  moment  of  inertia  needs  to  be  the  same  about  all  gravity  axes, 
and  this  of  course  points  to  a  circular  section.  Also  the  moment  of 
inertia  should  be  as  large  as  possible  for  a  given  weight  of  material, 
and  this  points  to  the  hollow  section.  The  disposition  of  the  metal 
in  a  circular  hollow  section  is  the  most  economical  one  for  long  col- 
umn machine  members  with  fixed  or  rounded  ends.  This  form, 
like  that  for  tension,  may  be  changed  to  the  rectangular  hollow  sec- 
tion if  appearance  requires  such  change.  If  the  long  column  ma- 
chine member  be  "  pin  connected,"  the  tendency  to  buckle  is  great- 
est in  a  plane  through  the  line  of  direction  of  the  compressive  force, 
and  at  right  angles  lo  the  axis  of  the  pins.  The  moment  of  inertia 
of  the  cross-section  should  therefore  be  greatest  about  a  gravity 
axis  parallel  to  the  axis  of  the  pins.  Example  :  a  steam  engine 
connecting-rod. 

When  the  machine  member  is  subjected  to  transverse  stress  the 
best  form  of  cross-section  is  probably  the  I  section,  a,  Fig.  153,  in 
which  a  relatively  large  moment  of  inertia,  with  economy  of  mater- 
ial, is  obtained  by  putting  the  excess  of  the  material  where  it  is 
most  effective  to  resist  flexure,  i.  e.,  at  the  greatest  distance  from  the 
given  gravity  axis.     Sometimes,  however,  if  the  I  section  has  to  be 


142  MACHINE    DESIGN. 

produced  by  cutting  away  the  material  ate  and  d,  in  the  machine 
shop,  instead  of  producing  the  form  directly  in  the  rolls,  it  is 
cheaper  to  use  the  solid  rectangular  section  c.  If  the  member  sub- 
jected to  transverse  stress  is  for  any  reason  made  of  cast  material, 
as  is  often  the  case,  the  form  h  is  preferable,  for  the  following 
reasons  :  I.  The  best  material  is  almost  sure  to  be  in  the  thinnest 
part  of  a  casting,  and  therefore  in  this  case  is  at  /  and  g,  where  it 
is  most  effective  to  resist  flexure.  II.  The  pattern  for  the  form  b  is 
more  cheaply  produced  and  maintained  than  that  for  a.  III.  If 
the  surface  is  left  without  finishing  from  the  mould,  any  imperfec- 
tions due  to  the  foundry  work  are  more  easily  corrected  in  b  than 
in  a.  Machine  members  subjected  to  transverse  stress,  which  con- 
tinually change  their  position  relatively  to  the  force  that  produces 
the  flexure,  should  have  the  same  moment  of  inertia  about  all 
gravity  axes.  As,  for  instance,  rotating  shafts  that  are  strained 
transversely  by  the  force  due  to  the  weight  of  a  fly-wheel,  or  that 
due  to  the  tension  of  a  driving  belt.  The  best  form  of  cross-section 
in  this  case  is  circular.  The  hollow  section  would  give  the  greatest 
economy  of  material,  but  hollow  members  are  expensive  to  produce 
in  wrought  material,  such  as  is  almost  invariably  used  for  shafts. 
Hence  the  solid  circular  section  is  used. 

125.  Torsional  strength  and  stiffness  are  proportional  to  the  polar 
moment  of  inertia  of  the  cross-section  of  the  member.  This  is  equal 
to  the  sum  of  the  moments  of  inertia  about  two  gravity  axes  at 
right  angles  to  each  other.  The  forms  in  Fig.  153  are  therefore  not 
correct  forms  for  the  resistance  of  torsion.  The  circular  solid  or 
hollow  section,  or  the  rectangular  solid  or  hollow  section,  should  be 
used. 

The  I  section,  Fig.  154,  is  a  correct  form  for  resisting  the  stress 
P,  applied  as  shown.  Suppose  the  web  c  to  be  divided  on  the 
line  CD,  and  the  parts  to  be  moved  out  so  that  they  occupy  the 
positions  shown  at  a  and  b.  The  form  thus  obtained  is  called  a 
"  box  section."  By  making  this  change  the  moment  of  inertia 
about  AB  has  not  been  changed,  and  therefore  the  new  form  is  just 
as  effective  to  resist  flexure  due  to  the  force  P  as  it  was  before  the 


FORM    OP    MACHINE    PAliTS    AS    DICTATED    BY    STRESS.  143 

change.  The  hex  section  is  better  able  to  resist  torsional  stress, 
because  the  change  made  to  convert  the  I  section  into  the  box  sec- 
tion has  increased  the  polar  moment  of  inertia.  The  two  forms  are 
equally  good  to  resist  tensile  and  compressive  force  if  they  are 
sections  of  short  blocks.  But  if  they  are  both  sections  of  long 
columns,  the  box  section  would  be  preferable,  because  the  moments 
of  inertia  would  be  more  nearly  the  same  about  all  gravity  axes. 

126.  The  framing  of  machines  almost  always  sustains  com- 
bined stresses,  and  if  the  combination  of  stresses  include  torsion, 
flexure  in  different  planes,  or  long  column  compression,  the  box 
section  is  the  best  form.  In  fact  the  box  section  is  by  far  the  best 
form  for  the  resisting  of  stress  in  machine  frames.  There  are 
other  reasons,  too,  beside  the  resisting  of  stress  that  favor  its  use.* 
1,  Its  appearance  is  far  finer,  giving  an  idea  of  completeness  that  is 
always  wanting  in  the  ribbed  frames.  II.  The  faces  of  a  box  frame 
are  always  available  for  the  attachment  of  auxiliary  parts  without 
interfering  with  the  perfection  of  the  design.  III.  The  strength 
can  always  be  increased  by  decreasing  the  size  of  the  core,  without 
changing  the  external  appearance  of  the  frame,  and  therefore  with- 
out any  work  whatever  on  the  pattern  itself.  The  cost  of  patterns 
for  the  two  forms  is  probably  not  very  different ;  the  pattern  itself 
being  the  more  expensive  in  the  ribbed  form,  and  the  necessary  core 
boxes  adding  to  the  expense  in  the  case  of  the  box  form.  The 
expense  of  production  in  the  foundry,  however,  is  greater  for  the 
box  form  than  for  the  ribbed  form,  because  core  work  is  more  ex- 
pensive than  "  green  sand "  work.  The  balance  of  advantage  is 
very  greatly  in  favor  of  box  forms,  and  this  is  now  being  recognized 
in  the  practice  of  the  best  designers  of  machinery. 

127.  To  illustrate  the  application  of  the  box  form  to  machine 
members,  let  the  table  of  a  planer  be  considered.  The  cross-section 
is  almost  universally  of  the  form  shown  in  Fig.  155.  This  is  evi- 
dently a  form  that  would  yield  easily  to  a  force  tending  to  twist  it, 
or  to  a  force  acting  in  a  vertical  plane  tending  to  bend  it.  Such 
forces  may  be  brought  upon  it  by  "  strapping  down  work,"  or  by  the 

*  See  Richards'  "  Manual  of  Machine  Construction." 


144  MACHINE    DESIGN. 

support  of  heavy  pieces  upon  centres.  Thus  in  Fig.  156  the  heavy 
piece  E  is  supported  between  the  centres.  For  proper  support  the 
centres  need  to  be  screwed  in  with  a  considerable  force.  This 
causes  a  reaction  tending  to  separate  the  centres  and  to  bend  the 
table  between  C  and  D.  As  a  result  of  this,  the  Vs  on  the  table  no 
longer  have  a  bearing  throughout  the  entire  surface  of  the  guides  on 
the  bed,  but  only  touch  near  the  ends,  the  pressure  is  concentrated 
upon  small  surfaces,  the  lubricant  is  squeezed  out,  the  Vs  and 
guides  are  "  cut,"  and  the  planer  is  rendered  incapable  of  doing  ac- 
curate work.  If  the  table  were  made  of  the  box  form  shown  in 
Fig.  157,  with  partitions  at  intervals  throughout  its  length,  it 
would  be  far  more  capable  of  maintaining  its  accuracy  of  form  un- 
der all  kinds  of  stress,  and  would  be  more  satisfactory  for  the  pur- 
pose for  which  it  is  designed.* 

The  bed  of  a  planer  is  usually  of  the  form  shown  in  section  in 
Fig.  158,  the  side  members  being  connected  by  "cross  girts"  at  in- 
tervals. This  is  evidently  not  the  best  form  to  resist  flexure  and 
torsion,  and  a  planer  bed  may  sustain  both,  either  by  reason  of 
improper  support,  or  because  of  changes  in  the  form  of  foundation. 
If  the  bed  were  of  box  section  with  cross  partitions,  it  would  sus- 
tain greater  stress  without  undue  yielding.  Holes  could  be  left  in 
the  top  and  bottom  to  admit  of  supporting  the  core  in  the  mould  ; 
to  serve  for  the  removal  of  the  core  sand  ;  and  to  render  accessible 
the  gearing  and  other  mechanism  inside  of  the  bed. 

This  same  reasoning  applies  to  lathe  beds.  They  are  strained 
transversely  by  force  tending  to  separate  the  centres,  as  in  the  case 
of  "  chucking "  ;  torsionally  by  the  reaction  of  a  tool  cutting  the 
surface  of  a  piece  of  large  diameter  ;  and  both  torsion  and  flexure 
may  result,  as  in  the  case  of  the  planer  bed,  from  an  improperly 
designed  or  yielding  foundation.  The  box  form  would  be  the  best 
possible  form  for  a  lathe  bed  ;  some  difficulties  in  adaptation,  how- 
ever, have  prevented  its  extended  use  as  yet. 

These  examples  illustrate  principles  that  are  of  very  broad  ap- 
plication in  the  designing  of  machines. 

*Prof.  Sweet  has  designed  and  constructed  such  a  table  for  a  large  mill- 
ing machine. 


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FORM    OF    MACHINE    PARTS    AS    DICTATED    BY    STRESS.  145 

128.  Often  in  machines  there  is  a  part  that  projects  either  verti- 
cally or  horizontally  and  sustains  a  transverse  stress  ;  it  is  a  canti- 
lever, in  fact.  If  only  transverse  stress  is  sustained,  and  the  thick- 
ness is  uniform,  the  outline  for  economy  of  material  is  parabolic. 
In  such  a  case,  however,  the  outline  curve  of  the  member  should 
start  from  the  point  of  application  of  the  force,  and  not  from  the 
extreme  end  of  the  member,  as  in  the  latter  case  there  would  be  an 
excess  of  material.  Thus  in  A,  Fig.  159,  P  is  the  extreme  position 
at  which  the  force  can  be  applied.  The  parabolic  curve  a  is  drawn 
from  the  point  of  application  of  P.  The  end  of  the  member  is  sup- 
ported by  the  auxiliary  curve  c.  The  curve  h  drawn  from  the  end 
gives  an  excess  of  material.  The  curves  a  and  c  may  be  replaced 
by  a  single  continuous  curve  as  in  C,  or  a  tangent  may  be  drawn 
to  a  at  its  middle  point  as  in  B,  and  this  straight  line  used  for  the 
outline ;  the  excess  of  material  being  slight  in  both  cases.  Most 
of  the  machine  members  of  this  kind,  however,  are  subjected  also 
to  other  stresses.  Thus  the  "  housings "  of  planers  have  to  resist 
torsion  and  side  flexure.  They  are  very  often  supported  by  two 
members  of  parabolic  outline  ;  and,  to  insure  the  resistance  of  the 
torsion  and  side  flexure,  these  two  members  are  connected  at  their 
parabolic  edges  by  a  web  of  metal  that  really  converts  it  into  a  box 
form.  Machine  members  of  this  kind  may  also  be  supported  by  a 
brace,  as  in  D.  The  brace  is  a  compression  member  and  may  be 
stiffened  against  buckling  by  a  "web"  as  shown,  or  by  an  auxiliary 
brace. 


CHAPTE'R   XV. 

MACHINESUPPOETS. 

129.  The  Single  Box  Pillar  Support  is  best  and  simplest  for  ma- 
chines whose  size  and  form  admit  of  its  use.  When  a  support  is  a 
single  continuous  member,  its  design  should  be  governed  by  the 
following  principles: 

I.  The  amount  of  material  in  the  cross-section  is  determined  by 
the  intensity  of  the  load.  If  vibrations  are  also  to  be  sustained, 
the  amount  of  njaterial  must  be  increased  for  this  purpose. 

II.  The  vertical  centre  line  of  the  support  should  coincide  with 
the  vertical  line  through  the  centre  of  gravity  of  the  part  supported. 

III.  The  vertical  outlines  of  the  support  should  taper  slightly 
and  uniformly  on  all  sides.  If  they  were  parallel  they  would 
appear  nearer  together  at  the  bottom. 

IV.  The  external  dimensions  of  the  support  must  be  such  that 
the  machine  has  the  appearance  of  being  in  stable  equilibrium. 
The  outline  of  all  heavy  members  of  the  machine  supported  must 
be  either  carried  without  break  to  the  foundation,  or  if  they  over- 
hang, must  be  joined  to  the  support  by  means  of  parabolic  outlines, 
or  by  the  straight  lines  of  the  brace  form. 

Illustration.  —  In  Fig.  160  the  first  three  principles  may  be  ful- 
filled, but  there  is  an  appearance  of  instability.  It  is  because  the 
outline  of  the  "  housing "  overhangs.  It  should  be  carried  to  the 
foundation  without  break  in  the  continuity  of  the  metal,  as  in 
Fig.  161. 

130.  AVhen  the  support  is  divided  up  into  several  parts,  modi- 
fication of  these  principles  becomes  necessary,  as  the  divisions 
require  separate  treatment.     This  question   may  be  illustrated  by 


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MACHINE    SUPPORTS.  147 

lathe  supports.  In  Fig.  162  are  shown  three  forms  of  suppoi-t  for  a 
lathe,  seen  from  the  end.  For  stability  the  base  needs  to  be  broader 
than  the  bed.  In  A  the  width  of  base  necessary  is  determined  and 
the  outlines  are  straight  lines.  The  unnecessary  material  is  cut 
away  on  the  inside,  leaving  legs,  which  are  compression  members  of 
correct  form.  The  cross  brace  is  left  to  check  any  tendency  to 
buckle.  For  convenience  to  the  workmen  it  is  desirable  to  naiffow 
this  support  somewhat  without  narrowing  the  base.  The  cross 
brace  converts  the  single  compressson  member  into  two  compression 
members.  It  is  allowable  to  give  these  different  angles  with  the 
vertical.  This  is  done  in  B  and  the  straight  lines  are  blended  into 
each  other  by  a  curve.  C  shows  a  common  incorrect  form  of  lathe 
support,  the  compression  members  from  the  cross  brace  downward 
being  curved.  There  is  no  reason  for  this  curved  form.  It  is  less 
capable  of  bearing  its  compressive  load  than  if  it  were  straight,  and 
is  no  more  stable  than  the  form  6,  the  width  of  base  being  the  same: 

Consider  the  lathe  supports  from  the  front.  Four  forms  are  shown 
in  Fig.  163.  If  there  were  any  force  tending  to  move  the  bed  of  the 
lathe  endwise  the  forms  B  and  C  would  be  allowable.  But  there  is 
no  force  of  this  kind,  and  the  correct  form  is  the  Otie  shown  in  D. 
Carrying  the  foot  out  as  in  A,  B,  and  C,  increases  the  dista,nce 
between  supports  (the  bed  being  a  beam  with  end  supports  and  the 
load  between);  this  increases  the  deflection  and  the  fibre  stress  due 
to  the  load.  This  increase  in  stress  is  probably  not  of  any  serious 
importance,  but  the  principle  should  be  regarded  or  the  appearance 
of  the  machine  will  not  be  right.  If  the  supports  were  joined  by  a 
cross  member,  as  in  Fig.  164,  they  would  be  virtually  converted  into 
a  single  support,  and  should  then  taper  from  all  sides. 

131.  If  a  machine  be  supported  on  a  single  box  pillar,  change^  in 
the  form  of  the  foundation  cannot  induce  stress  in  the  machine 
frame  tending  to  change  its  form.  If,  however,  the  machine  is  sup- 
ported on  four  or  more  legs  the  foundation  might  sink  away  from 
one  or  more  of  them  and  leave  a  part  unsupported.  This  might 
cause  torsional  or  flexure  stress  in  some  part  of  the  machine,  which 
might  change  its  form,  and  interfere  with  the  accuracy  of  its  action. 


148  MACHINE    DESIGN. 

But  if  the  machine  he  supported  on  three  points  this  cannot  occiii, 
V)ecause,  if  the  foundation  should  sink  under  any  one  of  the  sup- 
))()rt8,  the  support  would  follow  and  the  machine  would  still  rest  on 
three  points.  When  it  is  possible,  therefore,  a  machine  which  can- 
not be  carried  on  a  single  pillar  should  be  supported  on  three 
points.  Many  machines  are  too  large  for  three-point  support,  and 
the  resource  is  to  make  the  bed,  or  part  supported,  of  box  section 
and  so  rigid  that  even  if  some  of  the  legs  should  be  left  without 
foundation,  the  part  supported  would  still  maintain  its  form.  More 
supports  are  often  used  than  are  necessary.  Thus,  if  a  lathe  has 
two  pairs  of  legs  like  those  shown  in  B,  Fig.  162,  and  these  are  bolted 
firmly  to  the  bed,  there  will  be  four  points  of  support.  But  if,  as 
suggested  by  Professor  Sweet,  one  of  these  pairs  be  connected  to  the 
bed  by  a  pin  so  that  the  support  and  the  bed  are  free  to  move, 
relatively  to  each  other,  about  the  pin,  as  in  Fig.  165,  then  this  is 
equivalent  to  a  single  support,  and  the  bed  will  have  three  points  of 
support,  and  will  maintain  its  form  independently  of  any  change 
in  the  foundation.  This  is  of  special  importance  when  the  ma- 
chines are  to  be  placed  upon  yielding  floors. 

132.  Fig.  166  shows  another  case  in  which  the  number  of  sup- 
])ort8  may  be  reduced  without  sacrifice.  In  A  three  pairs  of  legs 
are  used.  There  are  therefore  six  points  of  support.  In  B  two 
pairs  of  legs  are  used  and  one  may  be  connected  by  a  pin,  and  there 
will  be  but  three  points  of  support.  The  chance  of  the  bed  being 
strained  from  changing  foundation,  has  been  reduced  from  6  in  ^ 
to  0  in  B.  The  total  length  of  bed  is  12  ft.,  and  the  unsupported 
length  is  6  ft.  in  both  cases. 

133.  Figs.  167  and  168  show  correct  methods  of  support  for  small 
lathes  and  planers,  due  to  Professor  Sweet.  In  Fig.  167  the  lathe 
"  head  stock  "  has  its  outlines  carried  to  the  foundation  by  the  box 
l»illar  ;  a  represents  a  pair  of  legs  connected  to  the  bed  by  a  pin 
connection,  and  instead  of  being  placed  at  the  end  of  the  bed  it  is 
moved  in  somewhat,  the  end  of  the  bed  being  carried  down  to  the 
support  by  a  parabolic  outline.  The  unsupported  length  of  bed  is 
thereby  decreased,  the  stress  on  the  bed  is  less,  and  the  bed  will 


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MACHINE   SUPPORTS.  149 

maintain  its  form  regardless  of  any  yielding  of  the  floor  or  founda- 
tion. In  Fig.  168  the  housings,  instead  of  resting  on  the  bed  as  is 
usual  in  small  planers,  are  carried  to  the  foundation,  forming  two 
of  the  supports  ;  the  other  is  at  a  and  has  a  pin  connection  with  the 
bed,  which  being  thus  supported  on  three  points  cannot  be  twisted 
or  flexed  by  a  yielding  foundation. 


CHAPTER   XVI. 

MACHINE     FRAMES. 

134.  Fig.  ^69  shows  an  open  side  frame,  such  as  is  used  for  punch- 
ing and  shearing  machines.  During  the  action  of  the  punch  or  shear 
a  force  is  applied  to  the  frame  tending  to  separate  the  jaws.  This 
force  may  be  represented  in  magnitude,  direction,  and  line  of  action 
by  P.  It  is  required  to  find  the  resulting  stresses  in  the  three  sec- 
tions AB^  CD,  and  EF.  Consider  AB.  Let  the  portion  above  this 
section  be  taken  as  a  free  body.  The  force  P,  Fig.  170,  and  the  op- 
posing resistances  to  deformation  of  the  material  at  the  section  AB, 
are  in  equilibrium.  Let  H  be  the  projection  of  the  gravity  axis  of 
the  section  AB,  perpendicular  to  the  paper.  Two  equal  and  oppo- 
site forces,  Pi  and  P.^,  may  be  applied  at  H  without  disturbing  the 
equilibrium.  Let  P^  and  P.^  be  each  equal  to  P,  and  let  their  line  of 
action  be  parallel  to  that  of  P.  The  free  body  is  now  subjected  to 
the  action  of  an  external  couple,  PI,  and  an  external  force,  P^.  The 
couple  produces  flexure  about  H,  and  the  force  P^  produces  tensile 
stress  in  the  section  AB.  The  flexure  results  in  a  tensile  stress 
varying  from  a  maximum  value  in  the  outer  fibre  at  ^  to  zero  at  H, 
and  a  compressive  stress  varying  from  a  maximum  in  the  outer 
fiber  at  B  to  zero  at  H.  This  may  be  shown  graphically  at  JK. 
The  ordinates  of  the  line  LM  represent  the  varying  stress  due  to 
flexure  ;  while  ordinates  between  LM  and  NO  represent  the  uni- 
form tensile  stress.  This  latter  diminishes  the  compressive  stress 
at  B,  and  increases  the  tensile  stress  at  A.  The  tensile  stress  per 
square  inch  at  A  therefore  equals  S  -\-  S^;  where  S  equals  the  unit 
fibre  stress  due  to  flexure  at  A,  and  S^  equals  the  unit  tensile  stress 

Pic  P 

due  to  P,.     Now  *S^  =  -r  .  and  S.  =  -.;   in  which  c  =  the  distance 
1  J  ,  'A' 


MACHINE    FRAMES.  151 

from  the  gravity  axis  to  the  outer  fibre  =  AH,  and  1=  the  mo- 
ment of  inertia  of  the  section  about  H,  and  A  =  area  of  the  cross- 
section  AB. 

Let  it  be  required  to  design  the  frame  of  a  machine  to  punch  |" 
holes  in  ^"  steel  plates,  18"  from  the  edge.  The  surface  resisting 
the  shearing  action  of  the  punch  =  tt  X  f"  X  V  =  1'17  sq.  in. 
The  ultimate  shearing  strength  of  the  material  is  say  50000  pounds 
per  square  inch.  The  total  force,  P,  which  must  be  resisted  by  the 
punch  frame  =  50000  X  1*17  =58500  pounds. 

135.  The  material  and  form  for  the  frame  must  first  be  selected. 
The  form  is  such  that  forged  material  is  excluded,  and  difficulties  of 
casting  and  high  cost  exclude  steel  casting.  The  material,  therefore, 
must  be  cast  iron.  Often  the  same  pattern  is  used  both  for  the 
frame  of  a  punch  and  shear.  In  the  latter  case  when  the  shear 
blade  begins  and  ends  its  cut  the  force  is  not  applied  in  the  middle 
plane  of  the  frame,  but  considerably  to  one  side,  and  a  torsional 
stress  results  in  the  frame.  Combined  torsion  and  flexure  are  best 
resisted  by  members  of  box  form.  The  frame  will  therefore  be 
made  of  cast  iron  and  of  box  section.  The  dimension  AB  may  be 
assumed  so  that  its  proportion  to  the  "  reach  "  of  the  punch  appears 
right ;  the  width  and  thickness  of  the  cross-section  may  also  be 
assumed.  From  these  data  the  maximum  stress  in  the  outer  fibre 
may  be  determined.  If  this  is  a  safe  value  for  the  material  used 
the  design  will  be  right. 

136.  Let  the  assumed  dimensions  be  as  shown  in  Fig.  171.    Then 

A  =  h^dy  —  h.^d.i  =  78  sq.  in. 

h^dl  —  h^ 
12 

^^  3000  bi-quadratic  inches,  nearly. 

c  —  c^j  -^-  2  —  9" ;  r=  the  reach  of  the  punch  -|-  c  =  27";  P  = 
58500  lbs.,  as  determined  above.     Then 


152  MACHINE   DESIGN. 


P      58500 


,  „      Pic      58500  X  27  X  9       ,„„„ 
^=.T  = 3000 =^^^- 

Si-\-  S  =  5630  =  maximum  stress  in  the  section. 

The  average  strength  of  cast  iron  such  as  is  used  for  machinery 
castings,  is  about  20000  lbs.  per  square  inch.  The  factor  of  safety 
in  the  case  assumed  equals  20000  -f-  5630  =  3*5.  This  is  too  small. 
There  are  two  reasons  why  a  large  factor  of  safety  should  be  used 
in  this  design:  I.  When  the  punch  goes  through  the  plate  the  yield- 
ing is  sudden  and  a  severe  stress  results.  This  stress  has  to  be  sus- 
tained by  the  frame,  which  for  other  reasons  is  made  of  unresilient 
material.  II.  Since  the  frame  is  of  cast  iron,  there  will  necessarily 
be  shrinkage  stresses  which  the  frame  must  sustain  in  addition  to 
the  stress  due  to  external  forces.  These  shrinkage  stresses  cannot 
be  calculated  and  therefore  can  only  be  provided  against  by  a  large 
factor  of  safety. 

Cast  iron  is  strong  to  resist  compression  and  weak  to  resist  ten- 
sion, and  the  maximum  fibre  stress  is  tension  on  the  inner  side. 
The  metal  can  therefore  be  more  satisfactorily  distributed  than  in 
the  assumed  section,  by  being  thickened  where  it  sustains  tension, 
as  at  a,  Fig.  172.  If,  however,  there  is  a  very  thick  body  of  metal 
at  a,  sponginess  and  excessive  shrinkage  would  result.  The  form 
B  would  be  better,  the  metal  being  arranged  for  proper  cooling  and 
for  the  resisting  of  flexure  stress. 

137.  Dimensions  may  be  assigned  to  a  section  like  B  and  the 
cross-section  may  be  checked  for  strength  as  before.  See  Fig.  173. 
GG,  a  line  through  the  centre  of  gravity  of  the  section,  is  found  to 
be  at  a  distance  of  7"  from  the  tension  side.  The  required  values 
are  as  follows  :  c  =  7" ;  I  =  reach  of  punch  -f  ^  =  18  -f  7  ^==  25"; 
A  =  156"5  sq.  in.;  /  =  5000  bi-quadratic  inches,  nearly;  P=  58500 
lbs. 


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MACHINE    FRAMES.  153 


Then  *^i  =  T  ^  -trn  -  =  '^74  lbs. 

_,       P?^'       58500X25X7       ,,^^7  ik 
^-~T- 5-000  =-2047  lbs. 

*S^,  +  /S  =  2421  lbs.  ==  maximum  fibre  stress  in  the  section.  The 
factor  of  safety  =  20000 -f- 2421  =8-25.  This  section,  therefore, 
fulfills  the  requirement  for  strength,  and  the  material  is  well 
arranged  for  cooling  with  little  shrinkage  and  without  spongy 
spots.  The  gravity  axis  may  be  located,  and  the  value  of  /  deter- 
mined by  graphic  methods.     See  Hoskins'  "Graphic  Statics." 

138.  Let  the  section  CD^  Fig.  169,  be  considered.  Fig.  174  shows 
the  part  at  the  left  of  CD  free.  K  is  the  projection  of  the  gravity 
axis  of  the  section.  As  before,  put  in  two  opposite  forces  P^  and  P^, 
equal  to  each  other  and  to  F,  and  having  their  common  line  of 
action  parallel  to  that  of  P,  at  a  distance  /,  irom  it.  F  and  P^  now 
form  a  couple,  whose  moment  ^^  Pl^^  tending  to  produce  flexure 
about  K.  P-i  must  be  resolved  into  two  components,  one  P.^  J,  at 
right  angles  to  the  section  considered,  tending  to  produce  tensile 
stress;  and  the  other  JK,  parallel  to  the  section,  tending  to  produce 
shearing  stress.  The  greatest  unit  tensile  stress  in  this  section  will 
equal  the  sum  of  that  due  to  flexure  and  that  due  to  tension  = 

IK 

The  greatest  unit  shear  =:      *S^^  =  *-j-. 

139.  In  the  section  FE,  Fig.  169,  which  is  parallel  to  the  line  of 
action  of  P,  equal  and  opposite  forces,  each  =  P,  may  be  introduced, 
as  P5  and  P^.  P  and  P^  will  then  form  a  couple  with  an  arm  l.^, 
and  Pg  will  be  wholly  applied  to  produce  shearing  stress.  The 
maximum  unit  tensile  stress  in  this  section  will  be  that  due  to 
flexure,  S  =  PL^c  -^-  /,  and  the  maximum  unit  shear  will  be  S^=^ 
P  -^  A.     Any  section  may  be  thus  checked. 


154  MACHINE    DESIGN. 

140.  The  dimensions  of  several  sections  being  found,  the  outline 
curve  bounding  them  should  bedrawn  carefully,  to  give  good  ap- 
pearance. The  necessary  modifications  of  the  frame  to  provide  for 
support,  and  for  the  constrainment  of  the  actuating  mechanism, 
may  be  worked  out  as  in  Fig.  175.  A  is  the  pinion  on  the  pulley 
shaft  from  which  the  power  is  received  ;  B  is  the  gear  on  the  main 
shaft ;  C,  D,  and  G  are  parts  of  the  frame  added  to  supply  bear- 
ings for  the  shafts  ;  E  furnishes  the  guiding  surfaces  for  the  punch 
"  slide."  Thie  method  of  supporting  the  frame  is  shown,  the  support 
being  cut  under  at /^^  for  convenience  to  the  workman.  The  parts 
C,  D,E,  and  (r  can  only  be  located  after  the  mechanism  train  has 
been  designed. 

141.  Slotting  Machine  Frame.  —  See  Fig.  176,  It  is  specified  that 
the  Blotter  shall  ctit  at  a  certain  distance  from  the  edge  of  any 
piece,  and  the  dimension  AH  is  thus  determined.  The  table  G 
must  be  held  at  a  convenient  height  above  the  floor,  and  RK  must 
provide  for  the  required  range  of  "  feed."  K  is  cut  under  for  con- 
venience to  the  workman,  and  carried  to  the  floor  line  as  shown. 
It  is  required  to  "slot "  a  piece  of  given  vertical  dimension,  and  the 
distance  from  the  surface  of  the  table  to  E  is  thus  determined.  Let 
the  dimension  LM  be  assumed  so  that  it  shall  be  in  proper  propor- 
tion to  the  necessary  length  and  height  of  the  machine.  The  curves 
LS  and  MT  may  be  drawn  for  bounding  lines  of  a  box  frame  to 
support  the  mechanism.  M  should  be  carried  to  the  floor  line  as 
shown,  and  not  cut  under.  None  of  the  part  DNE,  nor  that  which 
serves  to  support  the  cone  and  gears  on  the  other  side  of  the  frame, 
should  be  made  flush  with  the  surface  LSTM,  because  nothing 
should  interfere  with  the  continuity  of  the  curves  LS  and  TM.  The 
supporting  frame  of  a  machine  should  he  clearly  outlined,  and  other 
parts  should  appear  as  attachments.  The  member  VW  should  be 
designed  so  that  its  inner  outline  is  nearly  parallel  to  the  outline  of 
the  cone  pulley,  and  should  be  joined  to  the  main  frame  by  a  curve. 
The  outer  outline  should  be  such  that  the  width  of  the  member 
increases  slightly  from  W  to  F,  and  should  also  be  joined  to  the 
main  frame  by  a  curved  outline.     In  any  cross-section  of  the  frame, 


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TJBI"^ 


MACHINE   FRAMES.  155 

as  XX,  the  amount  of  metal  and  its  arrangement  may  be  controlled 
by  the  core.  It  is  dictated  by  the  maximum  force,  /*,  which  the 
tool  can  be  required  to  sustain.  The  tool  is  carried  by  the  slider  of 
a  slider  crank  chain.  Its  velocity  varies,  therefore,  from  a  max- 
imum near  mid-stroke,  to  zero  at  the  upper  and  lower  ends  of  its 
stroke.  The  belt  which  actuates  the  mechanism  runs  on  one  of  the 
steps  of  the  cone  pulley,  at  a  constant  velocity.  Suppose  that  the 
tool  is  set  (accidentally)  so  that  it  strikes  the  table  just  before  the 
slider  has  reached  the  lower  end  of  its  stroke.  The  resistance,  jR, 
offered  by  the  tool  to  being  stopped,  multiplied  by  its  (very  small) 
^velocity,  equals  the  difference  of  belt  tension  multiplied  by  the  belt 
velocity  (friction  and  inertia  neglected),  r  i2,  therefore,  would  vary 
inversely  as  the  slider  velocity,  and  hence  may  be  very  great.  Its 
maximum  value  is  indeterminate.  A  "breaking  piece  "  may  be 
put  in  between  the  tool  and  the  crank.  Then  when  R  reaches  a 
certain  value,  the  breaking  piece  fails.  The  stress  in  the  stress- 
members  of  the  machine  is  thereby  limited  to  a  certain  definite 
value.  From  this  valuie  the  frame  may  be  designed.  Let  P=  up- 
ward force  against  the  tool  when  the  breaking  piece  fails.  Let 
I  =  the  horizontal  distance  from  the  line  of  action  P  to  the  gravity 
axis  of  the  section  XX.  Then  the  section  XX  sustains  flexure  stress 
caused  by  the  moment  PI,  and  tensile  stress  equal  to  P.  The  max- 
imum unit  stress  in  the  section  = 

'  Pir  P 

s  +  s,  =  ^'  +  ^. 

A  section  may  be  assumed  and  checked  for  safety,  as  for  the  punch. 
142.  Stresses  in  the  Frame  of  a  Side-Crank  Steam  Engine.  —  Fig. 
177  is  a  sketch  in  plan  of  a  side-crank  engine  of  the  "  girder  bed  " 
type.  The  supports  are  under  the  cylinder  C,  the  main  bearing  E, 
and  the  out-board  bearing  D.  A  force  P  is  applied  in  the  centre  line 
of  the  cylinder,  and  acts  alternately  toward  the  right  and  toward 
the  left.  In  the  first  case  it  tends  to  separate  the  cylinder  and 
main  shaft ;  and  in  the  second  case  it  tends  to  bring  them  nearer 


156  MACHINE    DESIGN. 

together.  The  frame  resists  these  tendencies  with  resulting  inter- 
nal stresses. 

Let  the  stresses  in  the  section  AB  be  considered.  The  end  of 
the  frame  is  shown  enlarged  in  Fig.  178.  If  the  pressure  from  the 
piston .  is  toward  the  right,  the  stresses  in  AB  will  be  :  I.  Flexure 
due  to  the  moment  PI,  resulting  in  tensile  stress  below  the  gravity 
axis  N,  with  a  maximum  value  a;t  6,'^ tid' a' compressive  stress  above 
N  with  a  maximum  value  at  a.  II.  A  direct  tensile  stress,  =  P, 
distributed  over  the  entire  section,  resulting  in  a  unit  stress  = 
P  ~  A  =  »S  Jbs.  per  sq.  in.  This  is  shown  graphically  at  n,  Fig.  178. 
a,6j  is  a  datum  line  whose  length  equals  AB.  Tensions  are  laid  off 
toward  the  right  and  compressions  toward  the  left.  The  stress  due 
to  flexure  varies  directly  as  the  distance  from  the  neutral  axis  N^, 
being  zero  at  N^.  If,  therefore,  h^c^  represents  the  tensile  stress  in 
the  outer  fibre,  then  c^k\  drawn  through  N  will  be  the  locus  of  the 
ends  of  horizontal  lines,  drawn  through  all  points  of  a^h^,  represent- 
ing the  intensity  of  stress,  in  all  parts  of  the  section,  due  to  flexure. 
If  Cid^  represent  the  unit  stress  due  to  direct  tension,  then,  since  this 
is  the  same  in  all  parts  of  the  section,  it  will  be  represented  by  the 
horizontal  distance  between  the  parallel  lines  cjc^  and  d-^ey  This 
uniform  tension  increases  the  tension  h^c^  due  to  flexure,  causing 
it  to  become  bid^ ;  and  reduces  the  compression  k^a^,  causing  it 
to  become  e^a^.  The  maximum  stress  in  the  section  is  therefore 
tensile  stress  in  the  lower  outer  fibre,  and  is  equal  to  h^d^. 

When  the  force  P  is  reversed,  acting  toward  the  left,  the  stresses 
in  the  section  are  as  shown  at  m  :  compression  due  to  flexure  in  the 
lower  outer  fibre  equal  to  cfi.^ ;  tension  due  to  flexure  in  the  upper 
outer  fibre  equal  to  a.^k.^ ;  and  uniform  compression  over  the  en- 
tire surface  equal  to  d.^c.^.  This  latter  increases  the  compression 
in  the  lower  outer  fibre  from  h./,.^  to  h.^d.^,  and  decreases  the  tension 
in  the  upper  outer  fibre  from  a.,k.^  to  a./.^.  The  maximum  stress  in 
the  section  is  therefore  compression  in  the  lower  outer  fibre  equal  to 
h.^d.^.  The  maximum  stress,  therefore,  is  always  in  the  side  of  the 
frame  next  to  the  connecting-rod. 

If  the  gravity   axis  of  the  cross-section  be  moved  toward  the 


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MACHINE    FRAMES.  157 

connecting-rod,  the  stress  in  the  upper  outer  fibre  will  be  increased, 
and  that  in  the  lower  outer  fibre  will  be  proportionately  decreased. 
The  gravity  axis  may  be  moved  toward  the  connecting-rod  by 
Jncreasing  the  amount  of  material  in  the  lower  part  of  the  cross-^ 
section  and  decreasing  it  in  the  upper  part. 

The  stresses  in  any  other  section  nearer  the  cylinder  will  be  due 
to  the  same  force,  P,  as  before  ;  but  the  moment  tending  to  produce 
flexure  will  be  less,  because  the  lever  arm  of  the  moment  is  less  and 
the  force  constant. 

143.  Suppose  the  engine  frame  to  be  of  the  type  which  is  con- 
tinuous with  the  supporting  part  as  shown  in  Fig.  179.  Let  Fig. 
180  be  a  cross-section,  say  at  AB.  0  is  the  centre  of  the  cylinder. 
The  force  P  is  applied  at  this  point  perpendicular  to  the  paper.  C 
is  the  centre  of  gravity  of  the  section  (the  intersection  of  two 
gravity  axes  perpendicular  to  each  other,  found  graphically).  Join 
C  and  0,  and  through  C  draw  TX  perpendicular  to  CO.  Then  XX 
is  the  gravity  axis  about  which  flexure  will  occur.*  The  dangerous 
stress  will  be  at  F,  and  the  value  of  c  will  be  the  perpendicular  dis- 
tance from  i^to  XX.  The  moment  of  inertia  of  the  cross-section 
about  XX  may  be  found,  =^  /  ;  I,  the  lever  arm  of  P,  =  OC.  The 
stress  at  Fy  S  -f-  S^,  must  be  safe  value. 

S  =  -j^,  in  known  terms^ 

P 

S,  = :r^ — r-,  in  knowH  terms. 


area  of  sec'n' 


*This  is  not  strictly  true.  If  OC  is  a  diameter  of  the  **  ellipse  of  inertia," 
flexure  will  occur  about  its  conjugate  diameter.  If  the  section  of  the  engine 
frame  is  symmetrical  with  respect  to  a  vertical  axis,  OC  is  vertical,  and  its 
conjugate  diameter  XX  is  horizontal.  Flexure  would  occur  about  XX,  and 
the  angle  between  OC  and  XX  would  equal  90°.  As  the  section  departs  from 
symmetry  about  a  vertical,  XX,  at  right  angles  to  OC,  departs  from  OC's  con- 
jugate, and  hence  does  not  represent  the  axis  about  which  flexure  occurs. 
In  sections  like  Fig.  178,  the  error  from  making  13  —  90"  is  unimportant. 
When  the  departure  from  symmetry  is  very  great,  however,  OC's  conjugate 
should  be  located  and  used  as  the  axis  about  which  flexure  occurs.  For 
method  of  drawing  "  ellipse  of  inertia  "  see  Hoskins'  **  Graphic  Statics." 


158  MACHINE    DESIGN. 

144.  Closed  Frames.— Fig.  181  shows  a  closed  frame.  The  mem- 
bers G  and  H  are  bolted  rigidly  to  a  cylinder  C  at  the  top,  and  to  a 
bed  plate,  D/),  at  the  bottom.  A  force  P  may  act  in  the  centre  line, 
either  to  separate  D  and  C,  or  to  bring  them  nearer  together.  The 
problem  is  to  design  (z,  if,  and  I)  for  strength.  If  the  three  mem- 
bers were  "pin  connected,"  see  Fig.  182,  the  reactions  of  C  upon 
A  and  B  at  the  pins  would  act  in  the  lines  EF  and  GH.  Then  if  P 
acts  to  bring  D  and  C  nearer  together,  compression  results  in  ^,  the 
line  of  action  being  EF\  compression  results  in  B,  the  line  of  action 
being  GH,  These  compressions  being  in  equilibrium  with  the  force 
P,  their  magnitude  may  be  found  by  the  triangle  of  forces.  From 
these  values  A  and  B  may  be  designed.  C  is  equivalent  to  a  beam 
whose  length  is  I,  supported  at  both  ends,  sustaining  a  transverse 
load  F,  and  tension  equal  to  the  horizontal  component  of  the  com- 
pression in  A  or  B.  The  data  for  its  design  would  therefore  be 
available.  Reversing  the  direction  of  P  reverses  the  stresses  ;  the 
compression  in  A  and  B  becomes  tension  ;  the  flexure  moment 
tends  to  bend  C  convex  downward  instead  of  upward,  and  the  ten- 
sion in  C  becomes  compression. 

145.  But  when  the  members  are  bolted  rigidly  together,  as  in 
Fig.  181,  the  lines  of  the  reactions  are  indeterminate.  Assump- 
tions must  therefore  be  made.  Suppose  that  G  is  attached  to  D  by 
bolts  at  E  and  A.  Suppose  the  bolts  to  have  worked  slightly  loose, 
and  that  P  tends  to  bring  C  and  D  nearer  together.  There  would 
be  a  tendency,  if  the  frame  yields  at  all,  to  relieve  pressure  at  E 
and  to  concentrate  it  at  A.  The  line  of  the  reaction  would  pass 
through  A  and  might  be  assumed  to  be  perpendicular  to  the  surface 
AE.  Suppose  that  P  is  reversed  and  that  the  bolts  at  A  are 
loosened,  while  those  at  E  are  tight.  The  line  of  the  reaction 
would  pass  through  E,  and  might  be  assumed  to  be  perpendicular 
to  EA.  MN  is  therefore  the  assumed  line  of  the  reaction,  and  the 
intensity  =  7^  -^  2.  In  any  section  of  G,  as  XX^  let  K  be  the  pro- 
jection of  the  gravity  axis.  Introduce  at  K^  two  equal  and  opposite 
forces,  equal  to  R  and  with  their  lines  of  action  parallel  to  that  of 
R.     Then  in  the  section  there  is  flexure  stress  due  to  the  flexure 


MACHINE    FRAMES.  159 

moment  Rl,  and  tensile  stress  due  to  the  component  of  R.^  perpen- 
dicular to  the  section,  =  R^.  Then  the  maximum  stress  in  the  sec- 
tion =  S  f  S,. 

A  section  may  be  assumed,  and  A,  I,  and  c  become  known  ; 
the  maximum  stress  also  becomes  known,  and  may  be  compared 
with  the  ultimate  strength  of  the  material  used. 

Obviously  this  resulting  maximum  stress  is  greater  when  the 
line  of  the  reaction  is  MN,  than  when  it  is  KL.  Also  it  is  greater 
when  MN  is  perpendicular  to  EA,  than  if  it  were  inclined  more 
toward  the  centre  line  of  the  frame.  The  assumptions  therefore 
give  safety.  If  the  force  P  could  only  act  downward,  as  in  a  steam 
hammer,  KL  would  be  used  as  the  line  of  the  reaction. 

146.  The  part  I)  in  the  bolted  frame,  is  not  equivalent  to  a  beam 

with  end  supports   and  a   central   load  like  C,  Fig.  182,  but  more 

nearly  a  beam  built  in  at  the  ends  with  central  load  ;  and  it  may  be 

so  considered,  letting  the  length  of  the  beam  equal  the  horizontal 

distance  from  E  to  E,  =  I.     Then  the  stress  in  the  mid-section  will 

PI  Pic 

be  due  to  the  flexure  moment  -^,  and  the  maximum  stress  =  *S^  =5  >. 

The  values  c  and  /  may  be  found  for  an  assumed  section,  and  S 
becomes  known. 


INDEX. 


Addendum,  148. 

Angularity  of  connecting-rod,  20. 

Annular  gears,  50. 

Belts,  75. 

centrifugal  force  of,  89. 

design  of,  81. 

transmission  by,  75. 
Bevel  gears,  59. 
Bolts  and  screws,  130. 

and  screws,  design  of,  131. 

cross-section  of,  to  resist  tension, 
135. 

elongation  of,  134. 

fastenings  to  hold   steam  cheat 
cover,  132. 

tendency  to  loosen  nuts,  136. 

Cams,  72. 

Cantilever  in  machines,  145. 
Centro  of  two  gears,  40. 
Centros  of  relative  motion,  13. 

in  compound  mechanism,  14. 
Complete  constrainmentof  motion,  5. 
Cone  pulleys,  78. 
Conservation  of  energy,  1, 
Constrained  motion,  3. 
Cotter,  138. 
Cycloidal  curves,  42. 

teeth,  60. 


Energy  in  machines,  28. 

is  transferred  in  time,  29. 

Feathers,  or  splines,  137. 

rules  for,  137. 
Fly-wheel,  pump,  97. 

steam  engine,  99. 

design  of,  93. 
Force  problems,  29. 

in  the  steam  engine,  35 
Form  of  machine  parts,  140. 

in  stress,   tension,    or    compres- 
sion, 140. 

in  transverse  stress,  141. 

in  torsional  stress,  142. 
Frame  of  machine,  143. 

slotting  machine,  154. 

closed,  158. 
Free  motion,  3. 
Function  of  machines,  2. 

Gears :  angular  velocity  ratio,  70. 
annular,  50. 
backlash,  48. 
bevel,  59. 

bevel,  design  of,  63. 
clearance,  48. 

compound  spur  gear  chains,  69. 
definitions,  45,  47. 
design  of  worm ,  67. 


162 


INDEX. 


Gears,  diametral  pitch,  48. 
face,  48. 
formulas,  55.' 
interchangeable,  51. 
interchangeable  involute,  53. 
laying  out,  54. 
skew  bevel,  64. 
solution  from  other  data,  68. 
strength  of  teeth,  56. 
total  depth,  48. 
tooth,  design  of,  57. 
spiral,  64. 
working  depth,  48. 
worm,  65. 

Generating  circle,  43. 
Guides,  128. 

Higher  pairs,  39. 

Independent    constrainment  of  mo- 
tion, 13. 

Instantaneous  motion  and  instanta- 
neous centres  or  centros,  7. 

Interchangeable  gears,  43. 

Involute  teeth,  61. 
tooth  outlines,  46. 

Journals,  design  of,  111. 
allowable  pressure,  112. 
bearings  and  boxes,  120. 
direction  of  motion,  112. 
frictional  resistance  of,  113. 
lubrication  of,  123. 
oiling,  124. 

pressure  in  thrust,  118. 
radiation  of,  114. 
stationary,  124. 
thrust,  117. 

Keys,  as  a  means  for  preventing  rela- 
tive rotation,  137. 


Keys,  rules  for,  138. 

Kinds  of  motion  in  machines,  6. 

Lever  crank  chain,  13. 
Linkages  or  motion  chains ;  mechan- 
isms, 11. 
Location  of  centros,  12. 
Loci  of  centros  or  centroids,  9. 

Machine  frames,  150. 
Machine  supports,  147. 
Machinery  of  application,  2. 
Machinery  of  transmission,  2. 
Means  for  preventing  relative  rota- 
tion, 137. 
Motion  independent  of  force,  6. 

Non-circular  wheels,  59. 

Open  frame  design,  152. 

side  frames,  150. 
Outline  of  machine  frame,  154. 

Pairs  of  motion  elements,  10. 

Parallel  or  straight  line  motions,  37. 

Passive  resistance,  3. 

Pitch  point,  41. 

Prime  mover,  2. 

Pulleys,  cone,  graphical  method,  79. 

Racks,  48. 

Rate  of  doing  work,  28. 

Ratio  of  a  quick  return,  21. 

Reduction  of    the    number  of  sup- 
ports, 148. 

Relative    linear     velocity     in    same 
link,  17. 
linear     velocity     not     in     same 

Unk,  18. 
motion,  6. 

Rigid  body,  7. 

Riveted  joints,  100. 


INDEX. 


163 


Riveted  joints,  butt,  101. 
lap,  101. 

margin  of  rivets,  106. 
pitch  of  rivets,  105. 
table  for,  103. 

table  for  rivet  diameters,  104. 
table  of  efficiency,  107. 

Shrink  and  force  fits,  139. 
Slide?  crank  chain,  12. 

mechanism,  12. 

and  guide  of  unequal  length,  12'i 
Sliding  surfaces,  126. 
Slotted  cross-head  mechanism,  14. 
Slotting  machine  frame,  154. 
Solution  of  a  quick  return,  23. 
Steam  engine,  stresses  in,  155. 


Stresses  in  the  frame  of  a  steam  en- 
gine, 155. 
Support  divided  into  several    parts, 
146. 
for  lathes,  148. 
single  box  pillar,  147. 

Table  for  use  in  designing  belts,  85. 

Tooth  outlines,  41 

Toothed  wheels,  or  gears,  39. 

Vector,  16. 
Velocity,  15. 

of  cutting  tools,  21. 

Whitworth  quick  return  mechanism, 
25. 


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